# Full text of "An experimental enquiry concerning the natural powers of water and wind to turn mills"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's hooks while helping authors ami publishers reach new audiences. You can search through I lie lull text of this book on I lie web at |http : //books . qooqle . com/| 7*3" 153 ^/Muv ^ te-ivubd { A fate.. ti^.^ct{- S-*/ ?A itj. -uZa-£ •*>.„* ^\ r \L Experimental Enquiry CONCERNING THE NATURAL POWERS O F WATER and WIND TO Turn MILLS, and other Machines, depending on a circular Motion. 10 AN W By j/SMEATON, F.R.S. 1 4-n^v \ ii ' 7"- LONDON: Printed in the Year M.dcc.lx. v *- U»l^ A N Experimental ENQUIRY CONCERNING THE NATURAL POWERS O F W A T E R and WIND T O Turn MILLS, and other Machines, depending on a circular Motion. *«'"«n. TITHAT I have to communicate on *■«, .759- yy this fubject was originally deduced frora experiments made on working models, which I look upon as the beft means of obtaining the out- lines in mechanical enquiries. . But in this cafe it is very neceuary to diJiinguHh the circumftances in which a model diners from a machine in large; otherwife a model is more apt to lead us from the truth than towards it. Hence the common obferva- A a tion, 181562 • " " E 4 •} • ^ ■ ■ tlon, that a thing may do very well in a model,, that will not anfwer in large. And indeed, tho' the ut- moft circumfpeftion be ufed in this way, the beft ftrudture of machines cannot be fully afcertained,. but by making trials With them, when made of their proper fize. It is for this reafon, that, tho' the mo- dels referred to, and the greateft part of the follow- ing experiments, were made in the years 1772 andr 1 7 f 3 , yet I deferred offering them: to the Society; till I had an opportunity of putting the deductions made therefrom in real practice, in a variety of cafes* and for various purpofes ; fo as to be able to aflure: the Society, that I have found them to anfwer. . PARt I. Concerning Undershot Water-Wheels. Plate IV. Fig. 1. is a perfpedive view of the ma- chine for experiments on water-wheels ; wherein A BCD is the lower cittern, or magazine, for re- ceiving the water, after it has quitted the wheel ;, . and for fupplying D E the upper cifteni, or head 5 wherein the water being railed to any height required, by a pump, that height is fliewn by F G, a fmall rod, divided into inches and parts j. with a float at the bottom, to move the rod up and down, as the furface of the water riles and' falls. . H I is a rod by which the fluice is drawn, and ftopt at any height required, by means of K a pin or peg, which fits feveral holes, placed in [ s ] In the manner of a diagonal (bale, upon the face of the rod HI.. G L is the upper part of the rod of the pump, for drawing the water out of the lower ciftern, in order to raife and keep up the furface thereof at its defired height, in the head DEj thereby to fupply the water, expended by the aperture of the fluke. M M is the arch and handle for working the pump, which is limited in its ftroke by N a piece for flopping the handle from railing the pifton too high $ that alfo being prevented from going too low, by meeting the bottom bf the barrel. O is the cylinder, upon which a cord winds, and which being conducted over the pulliea P and Q^ raifes - R, the fcale, into which the weights are put, for trying the power of the water. ,S T the two ftandards, which fupport the wheel, are made to Aide up and down, in order to ad- juft the wheel, as near as poffible, to the floor of the conduit; W the beam which fupports the fcale and pul- lies ; this is reprefented as but little higher than the machine, for the fake of bringing the figure into a moderate compafs, but in reality is placed I f or i & feet higher than the wheel. Plate V. Fig. a. is a fe&ion of the fame machine, wherein the fame parts are marked with the fame letters as in Fig. i. Befides which X X is the .pump barrel, being 5 inches diameter, and 1 1 inches long* Y is V _ 16} Y is the pifton ; and Z the fixed valve. GVija cylinder of wood, fixed upon the pump- rod, and reaches above the furface of the wa- ter ; this piece of wood being of fuch a thick- nefs, that its feftion is half the area of that of the pump-barrel, will caufe the furface of water to rife in the head, as much while the pifton is descending, as while it is rifing : and will there* by keep the gauge-rod F G more equally to its height. Note, the arch and handle M M is here reprefented on a different fide to what it is ihewn in the preceding figures, in order that its dim en - lions may the better appear. a a £bews one of the two wires which ferve as di- rectors to the float, in order that the gauge rod F G may be kept perpendicular ; for the fame purpofe alfo ferves w, a piece of wood with a hole to receive the gauge-rod, and keep it up- right. . h is the aperture of the fluice. c c a kant-board, for throwing the water more di- reftly down the opening c d, into the lower ciftern: and c e is a Hoping board, for bringing back the water that is thrown up by the floats of the wheel. Fig. $, represents one end of the main axis, with a fe&ion of the moveable cylinder, marked O in the preceding figures. ABCD is the end of the axis j whereof the parts B and D are covered with ferrules or hoops or brafs. £ is a cylinder of metal ; whereof the part marked Fis 8 [7l F is the pivot or gudgeon. cc is the feftion of an hollow cylinder of wood, the diameter of the interior part being fome- what larger than the cylindrical ferrule B. a a is the fe&ion of a ferrule of brafs, driven into the end of the hollow cylinder, and which is adjufted to that marked B, fo as to Hide freely thereupon, but with as little (hake as poffible. bb, dd,gg, reprefent the fe&ion of a brafs ferrule, plate, and focket, fixed upon the other end of the hollow cylinder} the focket dd being ad- jufted to Aide freely upon the cylinder E, in the iaroe manner as the ferrule *a Hides upon the . cylinder B : the outer end of the focket at g g is formed into a fort of button ; by pufhing wfeeneof^ the hollow cylinder will move back- wards and f oiwacds, or tarn round at pleafcre upon the cylindrical parts of the axis B and £• e * y i r, », icprefent the fefition of a brafe ferrule, al- fo fixed upon the hollow cylinder: the edge of -*e: i$ cut into teeth, in. the manner of k tcntratt wheel f and the edge thereof v * is cut in the manner of a ratchett* ' Of ccaifcqocnce, when the plate bddi h puflied clofe to the ferrule D, the teeth of the ferrule * e will lay ihctfd of G, a pin fixed into the axis ; by which means the hoflow cylinder as made to turn along with the wheel and axis:: but being drawn back by the button gg y the hollow cylinder is thereby, dif- ♦engaged .from the pin G, and ©eafes turning. JV&fr» The weight in the icale is prevented from [8] from running back, by a catch that plays in and lays hold of the ratchet < o<>. By this means the hollow cylinder upon which the cord winds, and raifes the weight, is put in ad: ion and difcharged therefrom inftantaneoufly, while the wheel is in motion : for without fome contrivance of this kind, it would not be eafy to make this fort of experiments with any to- lerable degree of exa&nefs; • The ufe of the . apparatus now defcribed will be rendered more intelligible, by giving a general idea of what I had in view ; but as I fhall be obliged to make ufe of a term which has heretofore been the caufe of difputation, I think it neceffary to affign the fenfe in which I would be uuderftood to ufe it ; and in which I apprehend it is ufed by practical Mecha- nic ks. The word Power? as ufed in pradical mechanicks, I apprehend to fignify the exertion of ftrength, , gra- vitation, impulfe, or prefTure, fo as to produce mo- tion: and by means of ftrength, gravitation, im- pulfe, or preflure, compounded with motion, to be capable of producing an effedt : and that no effect is properly mechanical, but what requires fuch a kind of power to produce it. The railing of a weight, relative to the height to which it can be raifed in a given time, is the moil proper meafure of power ; or, in other words, if the weight raifed is multiplied by the height to which it can be raifed in a given time, the produdt is the meafure of the power railing it ; and confequently, all thofe powers are equal, whofe products, made by fuch / [ 9 ] foch multiplication, are equal: for if a power can raiie twice the weight to the fame height ; or the Tame weight to twice the height, in the fame time that another power can, the firft power is double the fecond : and if a power can raife half the weight to double the height j or double the weight to half the height, in the fame time that another can, thofe two powers are equal. But note, all this is to be under- ftood in cafe of flow or equable motion of the body raifed ; for in quick, accelerated, or retarded mo- tions, the vis inertia of the matter moved will make a variation. In comparing . the effects produced by water- wheels, with the powers producing them ; or, in other words, to know, what part of the original power is neceffarily loft in the application, we mull prev&oufly know now much of the power is fpent in overcoming the fridion of the machinery, and the refiftance of the air $ alfo what is the real velo- city of the water at the inftant that it ftrikes the wheel ; and the real quantity of water expended in a given time. , From the velocity of the water, at the inftant that it ftrikes the wheel, given ; the height of head pro- ductive of fuch velocity can be deduced, from ac- knowleged and experimented principles of hydrofta- tics : fo that by multiplying the quantity, or weight of water, really expended in a given time, by the height of head fo obtained ; which muft be confi- dered as the height from which that weight of wa- ter had defcended in that given time j we ftiall fiave z product, equal to the original power of the water y "and clear of all uncertainty, that would arife from* the fri&iori of the water, in paffing fmall apertures ; B ; * r and [10] and from all doubts, arifing from the different fure of fpouting waters, afligned by different authors. On the other hand, the fum of the weights raifed by the a&ion of this water, and of the weight re- quired to overcome the fri&ion and refiftance of the machine, multiplied by the height to which the weight can be raifed in the time given, the product will be equal to the cfFed of that power ; and the proportion of the two products will be the proper-* tion of the power to the effefi : fo that by loading the wheel with different weights fuccefHycly, we ihall be able to determine at what particular load* and velocity of the wheel* the effeft is a maximum. The manner of finding the real velocity of the water, at the inftant of its ftriking the wheel ; the manner of finding the value of the fri&ion, refin- ance, &c. in any given cafe; and the manner of fading the real ex pence of water, fo far as cojok eerns the following experiments, without having re- course to theory ; being matters upon which, the fol- lowing determinations depend, it will be neceflary to explain them* To determine the Velocity of the Water Jhrihkng the Wbeeh It has already been mentiened^ in the reference* ta the figures* that weighta are raifed by a cord winding round a cylindrical part of the ax$. Firft, then, let the wheel be put in motion by the water, but without any weights in the fcale h and let the number of turns in a minute be 60 1 now it is evi ? <knty that was the wheel free from fridion and refin- ance,, that 60. times the circumference of the wheel :• •: ....:: Would • » [» 3. would be the fpace through which the water would have moved in a minute * with that velocity where- with it ftruck the wheel: but the wheel being in- cumbred by fri&toti and refinance, and yet moving 60 turns in a minute, it is plain, that the velocity of the water muft have been greater than 60 cir- cumferences before it met with the wheel. Let now the cord be wound round the cylinder, but contrary to the ufual way, and put a weight in the fcale ; the weight fo difpofed (which may be called the counter-weight) will endeavour to affift the wheel in turning the fame away, as it would have been turned by the water : put therefore as much weight into the fcale as, without any water, will caufe it to turn fomewhat fafter than at the rate of 60 turns in a minute ; fuppofe 63 : let it now be tried again by the water, affifted by the weight ; the wheel therefore will now make more than 60 turns; fuppofe 64 : hence we conclude the water ftill ex- erts feme power in giving motion to the wheel. Let the wfeight be again increafed, fo as to make 6\\ turns in a minute without water : let it once more be tried with water as before ; and fuppofe it now to make the fame number of turns with water as without, viz. 64I : hence it is evident, that in this cafe the wheel makes the fame number of turns in a minute, as it would do if the wheel had no fric- tion or refiftance at all j becaufe the weight is equi- valent thereto 5 for was it too little, the water would accelerate the wheel beyond the weight $ and if too great, retard it ; fo that the water now becomes a regulator of the wheel's motion ; and the velocity of its circumference becomes a meafure of the vclo* city of the water, B a In C " 3 In like manner, in feeking the greateft produfl; or maximum of effedt \ having found by trials what weight gives the greateft produdt, by limply multi- plying the weight in the fcale by the number of turns of the wheel, find what weight in the fcale, when the cord is on the contrary fide of the cylinder, will caufe the wheel to make the fame number of turn9 the fame way, without water $ it is evident that this weight will be nearly equal to all friction and refin- ance taken together; and confequently, that the weight in the fcale, with twice * the weight of the fcale, added to the back or counter- weight, will be equal to the weight that could have been raifed, fup- pofing the machine had been without friction or re- finance 5 and which multiplied by the height to which it was raifed, the product will be the greateft efFedt of that power. . The quantity of water expended is found thusr The pump made ufe of for replenishing the head, with water was fo carefully made, that no water efcaping back by the leathers,., it delivered the fame quantity of water at every ftroke, whether worked quick or flow ; and as the length of the ftroke was limited, confequently the value of one ftroke (or on account of more exadtnefs I a ftrokes) was known, by the height to which the water was thereby raifed in the head ; which being of a regular figure was eafily meafured. The fluice, by which the water was drawn upon the wheel, was made to flop at certain heights by a peg ; fo that when the peg was in the fame hole, * The weight of the fcale makes put of the weight both ways* 8 the r<3] the aperture . for . the effluent water was the fame. Hence the quantity of wates expended by any given head, and opening of the fluice, may be obtained : for by obferving how many ftrokes a minute was fuffi- cient to keep up the furface of the water at the given height, and multiplying the number of ftrokes by the value of each, the water expended by any given aperture and head in a given time will be given. Thefe things will be further illuftrated by going over the calculus of one fett of experiments. Specimen of a Sett of Experiments. The fluice drawn to the i ft hole. The water above the floor of the fluice 30 Inches. Strokes of the pump in a minute — 397 The head raifed by 12 ftrokes — 21 Inches. The wheel raifed the empty fcale, and made Qirns in a minute ■ ■ 80 With a counter- weight of 1 lb. 8 oz. it made 85 D° tri^d with water — — \ — 86 N° Weight. Turns' in a mm. Produft. . x — , 4 o — 45 — 180 2 — ' 5 6 — > 42 — 210 3, — 60 — 36% — 217! 4 — 7 o — 33* — 2361 5 — 8 o — 30— 240 maximum^ 6 — 9 o — 261 — 238$ 7 — 10 o — 22 — 220 8 — 11 o — 16} — 181* 9 — 12 * ceafed working. * N* B. When the wheel moves fo flbw as not to rid the wa- ter fo faft as fupplied by the fluice, the accumulated water falls back upon the aperture, and the wheal immediately ceafes moving. Counter C*4] Counter-weight, for 30 turns without water, a oz. in the fcale. •J/. B. The area of the head was 1 05,8 fqoare inches. Weight of the empty fcale and pulley, 10 oz. Circumference of the cylinder, 9 inches. Circumference of the water-wheel, 7$ ditto. • Reduction *f the above Sett tf Experiments. The circumference of the wheel, jf inches, mul- tiplied by 86 turns, gives 6450 inches for the velo- city of the water in a minute 5 ^~ of which will Be the velocity in a fecond, equal to 107,5 Inches, or 8,96 feet, which is due to a head of 1 f inches * ; and this we call the virtual or effe&ive head. The area of the head being 105,$ inches, this multiplied by the weight of Water of the inch cubic, equal to the decimal ,579 of the ounce avoirdupoife, gives 61,26 ounces for the weight of as much water, as is contained in the head, upon 1 inch in depth, tV of which is 3,83 pounds ; mis multiplied by the depth 2 1 inches, gives 80,43 lb. for the value of 1 2 ftrokes ; and by proportion, 3p| (the number made in a minute) will give 264^7 M?. the weight of wa- ter expended in a minute. _ Now as 264,7 lb* of water p&ay be coniidered as faviftg defended through a ipace of 1 5 inches in a minute, the prodoft of thefe two numbers 3970 will "exprefs the **w*r of the water to produce mechanical effects ; which were as fQllows. '^•IMMMVVf-WMW . * This, is det e rm ined upon the co mmon ma x im of h y d r o fl aticg» that the velocity of flouting; waters is equal to th* vilocky that an heavy body would acquire in falling from, the height of the refejVQir 1 and i* proved by the lifog ofjets to. the height of their vefervoift nearly. The The velocity of the wheel at the maximum, as ap- peals above, w&* 30 turns a minute ; which mul- tiplied bv$> inches, the circumference of the cylin- der, makes 270 inches $ but as the fcale was hung by a pulley and double line, the weight was only raifed half of this, viz. 135 inches. The weight in the fcale at the maximum $16. ooz. Weight of the fcale and pulley — — p 10 Counterweight, fcale, and pulley o 12 mm . Sum of the refiftance 9 6 ox lb. 9>57S* Now as 5>>375 ltx is raifed 13 J inches, thefe two numbers being multiplied together, the produd is 1266, which exprefles the eflfedt produced at a ma- ximum : fo that the proportion of the power to the effeB is as 3,970 : 1266, or as 10 : 3,18. But tho this is the greateft^g/i effect producible from the power mentioned, by the impulfe of the water trpts* aa andcrihot wheel ; yet, as the whole pernor of the water is not exhausted thereby, this will not be the true ratio between the fwkr of the water, and the ftm of all the effe&s producible thendfcooi. : for as the water muft acceflarily leave the wheel wkto a velocity equal to the wheel'* circum- ference, it is plain that fome part of the power of the water txxaSt remain after quitting, the wheel. The velocity of the wheel at the maximum is 30 turns a mhwte ; and confeqocntly its circumference moves at the rate of 7^1 2 5 feet a fecond, which an- fwers to a, head i,£i inches $ this bring multiplied by the expence of water in a minute, ofz. 2*64,7 lb; produces 481 for the power remaining in the water after it has pafled the wheel: this bemg therefore deduded [ i«] deduced from the original power 3970, leaves 3489, whidh is that part of the power which is fpent in producing the efFedt 1266; and confequently the part of the power fpent in producing the effect, is to the greateft efFe<3: producible thereby as 3489 : 1266 :: 10 : 3,62,or as 11 to 4. • The velocity of the water ftriking the wheel has been determined to be equal to 86 circumferences of the wheel per minute, and the velocity of the wheel at the maximum to be 30 $ the velocity of the water will therefore be to that of the wheel as 86 to 30, or as 10 to 3,5, or as 20 to 7. The load at the maximum has been fhown to be equal to 9 lb. 6 oz. and that the wheel ceafed move- ingwith 12 lb. in the fcale : to which if the weight of the fcale is added, viz. 10 ounces *, the propor- tion will be nearly as 3 to 4 between the load at the maximumznd that by which the" Wheel is flopped. It is fomewhat remarkable, that tho' the velocity of the wheel in relation to the water turns out greater than -j of the velocity of the the water, yet jthe im- pulfe of the water in the cafe of. a maximum is more than double of what is affigned by theory $ that is, inftead of | of the column, it is nearly ^qual to the whale column. It muft be remembred, therefore, that, in the pre- fent cafe, the wheel was not placed in an open river, where the natural current, after it has communicated its impulfe to the float, has room on all fides to es- cape, as the theory fuppofes ; but in a conduit or f * The refiftance of the air in this cafe ceafes, and the fri&ion is not added, as 12 lb. in the fcale was fufficient to flop the wheel after it had been in full motion ; and therefore fomewhat more than a counterbalance to the impulfe of the water, race, / E 17] race, to which thq fi^ftt }>c«p§ adapted, the water cannot other wife efcape than by moving along with {he wheel. It is obkfrvablq, that a wfieel working in this munnff, as foop as the water meets the float* receiving a fudd$n chepk, it fifes up agpinft (the flpat* like a w*ve ag^inft 3 fixed objeft $ infomuch that when the ihest <>f wader is not a quarter pf an ipcb thick before it pietts the fkgt, vet this Iheet wjll aft upon the whple furfaee of a float, iwhefe height is g inchqs ; said jconfequently was tibe flo^t qo higher than tto tfctckncjfs of die ihqet of waft?, as the cry alfo foppofes; a gteat j>art of the force wquid bare feet* lpft> hy tfae water's jdafhing over the £oat *. to fortfier confirmation of what is already deli- vofcd, J have joined the following tabl^ ,cQntaiiv* »g the *efu*k <n 27 fetts of (experiments, j^ade ^nd reduced in She ,pwtnj?e|r above ipeqiffcfl. ^h# jre*- rally 'fbflojW ftom a cqmpswjfon c£ the different ex- patimenfts tQgc&er, * Since ^he al)oye >vas ,v?r0te, J : find that Prpf<iflQr Eiiler^ip tfc Bedin A&s for thenar 1348^ in a jpemoire intitfied, fitaxims tour > which >fr cm to jbe; tie more reraariaUc, >as I qoat nnd ,bejha* >gbp*n4*y4)^0frtiation of the principle tfctrein contained* either &t^ ^ovy cfr-eKpetimertt; *>rthas mode ,atiy ufe thereof in h» 'ctffcmtottons cm this fiibj^b.— ^ dependant dans ce <faa puifque "** l^au eft r^ltechic, |& quelle fccoule Tur les allies vers Tes cqcEs, ■*" efie y exsree MCpre une force pa*tictt%e, flont r«$t de l^ni- ; " pulfionfeia aqgtxujite; &; experience jointer la tfcteorie a fait " 4C *oir jjue 4%»s .ce cas # Ja force sjfc .prefijue. double :..de ibue u qa'il Japt prendre le.4wi^le <te le fc&$a du til d'eaupour -oe M :i|ui r rfKp«nd ! dae3CQcas^a je furfeoe des aubes, pourvu.igu'fUe» *~ fci c nt a ffte largto pour rcc evoir ce fupplem e nt ^e forqe. Cfrfi ^ Its aube»t|ite>iem -plus targes que lefi], on t*ak d'eau on ne m4A ik vro it pieiidie qu e nc finale f e dj on, tQutcorrrrne dans le pre-^ M Aier^s, -on-Paube totfte entire eft pappee par Tesn^ C TABL J [ 18] TABLE I. IO:3,24 IO:3,2 'lO^,^ 10:3,02 IO.'2,85 IO:2,9 IO:2,8 10.'2,9 IO.'2,82 IO:3»075 IO:3,OI IO:2»92 IO:2,94 10:3,05 IO:2,98 10:3,4 10 : 3.5 !° : 3»4 10:3,45 10:3,36 10:3,6 ,Q :3^7 10:3,65 10:3,8 At the 10:7,75 i°;7>4 10:7,5 10:7,53 10; 7,32 10:8,02 lift 10:8,3 /bole, 10:9,1 10:9,1 10:9,3 ■— * ■ 22 12 23 24 9 6 25 26 I 27 '6 I. *• 10:3,23 10:3,05 10:3,01 1 o : 2,99 10:3,02 10:3,04 i° : 3.i3 10:3,06 10:3,62 10:3,6 10:3,62 10:3,97 10:4,1 ' f P ; 4>5 5 10:4,02 10:4,05 10:4,22 10:4,9 10:3,97 10:4,52 10:5,1 10:7,9 10.8,05 10:8,75 10:9, 10:8,7 [10:9,5 10:9, 10:3,03 4022,92 10:2,95 11* 10:4,55 10:4,9 10:5,2 IZi 10:9,17 10:9,5 10:9,35 10:9,45 10:9,3 10^9,25 6th. >3 8 Maxims r 19] r. Maxims and Obfervations deduced from the foregoing Table of Experiments. * Maxim I. That the virtual or effeftinje bead being the fame, the effeSt will be nearly as the quantity of water expended. This will appear by comparing the contents of the columns 4, 8, and io > in the foregoing fetts of ex- periments $ as for Example iji f taken from N°. 8. and 2$, viz; No. Virtual Head. Water expended.. • E0e& 8 7,29 — rr- l6i - — — r* 328 / a-f 7,29 355 — 785 Now the heads being equal $ if the eflecffs are pro- portioned to the water expended, we fhall have by maxim ift, 161 \ 357 :: 328 : 723* but 723 falls lhort of 78 57 as it turns out in experiment according to N°. 2 j, by 62 ; the effeft thertfofre of *J°. 25, compared with N°. 8, is greater tharr according to the prefent maxim in the ratio of 14 KrT3. ' : The foregoing example, wj^h four firnilarjqnes, are feen at one view m the following Tables : - , .. , f +—** 9 a vr ■Sr W o 4 I .. Comparifon. ife.{ H 2 '5 7>*9 .161. 355 "1 t8* iel > <« O f— « e CU <* Q O 5 *■» h O « 357:: 975:1221 «624» 4 H: 13 *r**t- -j: 8 1 0,5 10,5 285 357 I2IO| * J2U122 3d I 22 *1 6,8 6,8 2 SS Hi J86 fc** ; 3 * 2 :: 541 :7 ° 4 r8~ 3* : 391 «; 4»7 4»7 228 262 |g^|223:262:4 3i7t564 21+ 48-:.'! 7 *I3 5>°3 5»°3 307 360 S34 J 3 ° 7 : 36 ° : " 45 ° : 531 3 +| I78:l 77 C 2 Hence Hence therefore, in comparing different cxperi- Ifnerits, las fome fall ihorL and others exceed die maximum, and all agree therewith, as jiear as can fee expfe&ed, in tti affair where lb many different tii^flmttsftices are concerned 5 we may, according to the laws of reafoning by indu&ion, conclude the tnaifrft ttut 3 01 z. that ttie effefts are nearly as the '^uafitity df ^dttw: feXpflfded. M&kh A. 7&tf -ft&fr taprtir** <ef ' <wafer teing the fate, Hbe eJpS? «*»// 4i nearly *s *ke height *f the vif>m l vr effeai ik>beail. * » • • 'Thiis alio will appear by comparing the contents of 'columns 4, \ 2nd it>, la any of the fetts of ex- ^'mdrtts, : Jixaptfk •#> ^-W. a, and N°. 24. ««?. . • ,N\ V5tt.tkfd. E^«ce. £&&. -ft* — • *5 ■' — 864,7 " *a$6 24 _ 4,7 ^— . *6 a ,. „•,„„ , 3 .8f "Notir as&e^expewdis art taw quite equal, wemuft proportion one <* l!he fcfifccHs accordingly r 'thus Iby maxim ~Yft;~ a5a : 5264,7 :: '.jU j : £89 land by max. ad, jj : 4,7 :: n6o" : 397 Difference ^m , ■ 18 ie ie&<hqrefere of~N°. 4:4. oompaml with N°. •».4s lefs t han aec or din g-to the j ri fi^ 4U4xim ?a-tfap tiitoeff '4^:50. 'The fptegoing, and two -other JtmLiaregafOplcf, are comprised in the fallowing Table. I ** p •■■■» -Comiarifon. • f O cos 4 *H< «#* juy Ml Sr i*» OIL ..I 1266 1 Mpx. ift, 262 ^64,7 :: 385 ^ 319 7 a I , 385J i Mjix. 2d, a 5 : .4,7 :: 1266 5 3^1* f "lf v ' * }M«.4i y Jil-;^5 ;:ai7 4 ^1 ^. jl^x.2fti 5 & 53,5^: 14^:316 J *+ ■ 4 i » * — r— - r-s— f = — ; — j - \ »i — ■ ■ ■ — ; *iA 3f* i5<>5 IMf*- •* *^S;« 34* :: *42 : 433 1 . fi {MStf 6 h P} 2 fMfi*' & 'tf^'P? '5J5 : 45°f I 7 V s V 8 : 9 1 . ' * Ma^injL III. TJiaf the quality jof wafer txpendej km m g the fim* n the <$*& it ty(uJ)L,as tpe /qua re of .. This jviJJ appear by CQpiparinff the confcnts of ,^9lV!!mns : 3,^^iii fro, in : ^y,of *he Xetfs ©f ^xpcri. jgeots; as jfbr fiy^r |£ gf N*. * **/** **. 2,4. «fe. . ^T. Tuwis m a,mi«. -Expeiy*. Efieft. a < i w ftf .— * — p 2^4,7 -w-j — ia66 >4 .*- 1 — ^ $* .' i" y 2j62 -^--t — 38f ^e y^lcjcity^lng ;as Ae nmplber ojT turns, welhaljl type, ■■;■ " ^ " / . ;bynnx.4:ft, *$2 ,; 204,7 :: 38^ : 3*9 Tliflf*»ri>n/*» ••.•."' r TEhej^nVt gterejpse of N c . 24. compared with N°. -a.is-leis -than-by .the prefent maxim in _the.iatio jof 3?he foregoing, fand three : other jfira^ar examples, &p. fflmprSed in the following Table. ' . " Examples; [ * 2 ] Maxim ^ji I>3 3 • Maxim 4th. The. apir fun Being the fame, the effe3 witl'Ae mttrly as the -cube of the vekcityoftbe water. This alfo will appear by comparing the contents of columns 3, 8, and 10 ; as for Example 1/?, o/ : N\i,< and N°. 10, viz. N°. Turns, ' Expence. Effedh \ 1 ""„ " ■ ■ '"88 "r^ -27f — -— 141 1 10 ■ ■ ■ 42 — *- 114 — 117 Lemma. It muft here he dbferved, that if water paffes out of ab apdrture, in the fame fe&ioti, but with different > velocities $ the expence will bp pro- portional to the velocity; and therefore convierfely," if the expence is not proportional to the velocity, the fedtiori of the water is not the fame. Now comparing the water difcharged with the turns of N?. i. and 10, we fhall have 88 :54a :: 275 : 1 3 1, a ; but the water difcharged by N°. 10. is only 1 14 lb. therefore, tho' the fluice was drawn to the feme height in N ^ 10. *s in N°. 1. yet the fe&ion of the water palling out, wai.lefs in N°. 10. thanN°> i; in ihe proportion of 1*4 to 131,2 $ con- fequently had the effective aperture or fe&ion of the water been the fame inN c . 10. as in N°. 1. fothat 131,21b. of water had been difcharged irrftead of 114, the effedfc would have been increafed in. the fame proportion 4 that is, !by the Lemma, 88 : 42 : : 277 : 131,2 by maxim ift, 114 : 131,2 : : 117 : 134,5 andty max. 4 th , [ 68 ^;. 7 $b} s: I4M: J2H. Difference — — ip The . [Hi ■ Ute 4Fe& ttwtftfcre bf N°> ra compared #ith N°i i. is Ifefc than it bughi to be by to ptfMft maxim in the rati© of 7 : 8. The foregoing, aiifi Inree other ftmfoar examples,, are cbMai^a in me fcllotoihfr Tkble. > -*A H?UOI* . : -JodfOiJ •UOJJBIJEA J M CO . 8 4 T**^ c^ I & a? 3 \ «ft $ * ft *A ; & Sv4 ' C* ►* ^fr •• •• 1. «> c« c* u - •• CO CI IA 43% •• •• •• 00 « £ °° ^oo '- C« COCO *• •« ••■ •* fVt "* I" i. :' ». I ••• *• ••' ' Til ^S ** • • . • CO CO * CO CO O •Y V •• •• •• ^. c* '•■• • • •» ! 10 ¥h- I VO.^ M H* f O 00 T '« „ iv«06 ' Ji!- : /ajnuwu;* . *,* i v " in sftjnj. I l'<Hi&i : 00 ^ 1^ ^f ^-2 <. ^^ J Observa- Os ] Observations. Ob/erv. ift. Qn comparing column ad and 4th, Tab, I. it is evident, that the virtual bead bears ao certain proportion to the bead of water $ but that when the aperture is greater, or the velocity of the water iffuing therefrom lefe, they approach nearer to a coincidence : and confequently in the large open- ings of mills and flukes, where great quantities of water are difcharged from moderate heads, the head of water, and virtual head determined from the ve- locity, will nearly agree, as experience confirms. Obferv. 2d. Upon comparing the feveral pro* fxations between the pmier and effeB in column nth, the moft general is that of 10 to 3 $ the ex- tremes 10 to 3,2 and 10 to 2,8 5 but as it is obferv-* able, that where the .quantity of water, or the ve- locity thereof; that is, where the power is greateft, the 2d term of the ratio is greateft alfo : we may .therefore well allow the proportion fubfiftiog in large works, as 3*0 1. Obferv. 3*/. The proportions of velocities be- tween the water and wheel in column 1 2, are con- tained in the limits of 3 fa 1 and 2 to ij but as the greater velocities approach tfie limit of 3 to i # and the greater quantity of water approach to that of 2 to 1, the beft general proportion will be that of 5 to a. Ob fern. 4A&. On comparing the numbers 'in column *3» it appears, that there is no certain ratio .between the .load that the wheel will carry at its .«w,x/»w, and what will totally flop it; bijt tji^t .they are contained within the limits or 2Q to 10, and of 20 to i£ 5 but as the effe& approaches neareft to the ratio of 20 to 15, or of 4 to 3, when the power is greateft, whether by increafe of velocity, or quantity of water, this feems t9 be the moft appli- cable to large works : but as the load that a wheel ought to have, in order to work to the heft advan- tage, can be affigned, by knowing the effect it ought to produce, and the velocity it ought to have in pro- ducing it ; the exa<St knowlege of the greateft load it will bear, is of the lefs confequence in practice. It is to be rioted, that in all the examples under the three laft of the four preceding maxims, the effect of the lefler power falls ihort of its due proportion to the greater, when compared by its maxim ; except the laft example of maxim 4th : and hence, if the experiments are taken ftriftly, we muft infer, that the effects increafe and diminifh in an higher ratio than thofe maxims fuppofe : but as the deviation is not very confiderable, the greateft being about i*8th of the quantity in queftion ; and as it is not eafy to make experiments of fo compounded a nature with abfolute precifion ; we may rather fuppofe, that the lefler power is attended with fome friction, or works under fome difadvantage, which has not been duly accounted for, and therefore we may conclude, that, thefe maxims will hold very nearly, when applied to works in large. After the experiments above mentioned were tried, the wheel, which had originally 24 floats, was re- duced to twelve $ which caufed a diminution in the efleft, on account of a greater quantity of water cfcaping between the floats and the floor j but a cir- cular .1 [ *7 3 cular fweep being adapted thereto, of fuch a length, that one floit entered the curve before the preceding one quitted it, the effedt came fo near to the former, as not to give hopes of advancing it by increafing the number of floats beyond 24 in this particular wheel. P A R T II. Concerning Overshot Wheels, Read May 24, JN the former part of this eflay, we have - I 7S9- J[ confidered the impulfe of a confined jftream, adting on Under/hot Wheels. We now pro- ceed to examine the power and application of water, when adting by its gravity on Overjhot Wheels. , In reafoning without experiment, one might be led to imagine, that however different the mode of application is ; yet that whenever the fame quantity of water defcends thro' the fame perpendicular fpace, that the natural effedtive power would be equal; fuppofing, the machinery free from fridtion, equally calculated to receive the full effedt of the power, and to make the moft of it : for if we fuppofe the height of a column of water to be 30 inches, and refting upon a bafe or aperture of one inch fquare ; every cubic inch of water that departs therefrom will ac- quire the fame velocity or momentum,, from the uniform preffure of 30 cubic inches above it, that one cubic inch let fall from the top will acquire in /ailing down to the level of the aperture ; viz. fuch a velocity as in a contrary direction would carry it to Da the f «1 the level from whence it fell ; * one would therefore fuppofe, that a cubic inch of water, let fail thro* a fpace of 30 inches, and there impinging upon an- other body, would be capable of producing an equal effect by collifion, as if the fame cubic inch had de- fcended thro' the fame fpace with a flower motion,, and produced its effedts gradually : for in both cafes gravity adts upon an equal quantity of matter, thro* an equal fpace -f*; and confequently, that whatever was the ratio between the power and efFedt in under- fhot wheels, the fame would obtain in overfhot, and indeed in all others : yet, however conclufive this reafoning may feem, it will appear, in the courfe of the following deductions, that the efiedt of the gra- vity of defcending bodies is very different from the. effedl of the ftroke of fuch as are. hon-elafiic> tho* generated by an equal mechanical power. The alterations in the machinery already defGrrbedi. to accommodate the fame for experiments on over*- fhot wheels, were principally as follows^ Plate V. Fig. z* The fluice lb being (hut down, the rod HI was unfcrewed and taken off. The underftiot water-wheel was takeiuofF the axis,, and inflead thereof an overfhot wheel of the fame. * This is a confequence of the rifing of jctts to the height of' their refervoirs nearly. f Gravity, it is true, afls a longer fpace of time upon the body that defcends flow than upon that which falls quick ; but this can- not occafion the difference in the effed : for an elaftic body falling- thro* the fame fpace in the fame time, will, by collifion upon an- other elaftic body, rebound nearly to the height from which it fell 5, or, by communicating its motion, caufe an equal one to afcend to the fame height. diameter r*9j diariieter was put into its place- Note, This wheel was two inches in the fhroud or depth of the bucket $ the number of the buckets was 30 . The ftandards S and T, Fig. 1. were raifed half an inch, fo that the bottom of the wheel might be clear of ftagnant water. A trunk, for bringing the water upon the wheel r was fixed according to the dotted lines f g y Fig. 2. The aperture was adjufted by a {huttle b i y which alfo clofed up the outer end of the trunk, when the water was to be -ftoppecL Fig. 3. The ratchet v y riot being of one piece of metal with the ferrule e *, i i (tho' fo defcribed before, to prevent unneceffary diftindfions), was with, its catch turned the contrary fide } consequently the moveable barrel would do its office equally, notwith- ftanding the water- wheel, when at work, moved the contrary way- Specimen Specimen of a Sett of Experiments. Head 6 inches. a 4 \ ftrokes of the pump in a minute, ia ditto =s 80 ib. * Weight of the fcale (being wet) io£ oz. Counterweight for ao turns, befides the fcale, 3 oz. Weight in No. the Scale. Turns. Produ&. Obfervations. 1 — — olb. — 60 — — 1 Threw moft part 2 - ■ ■ ■ . 1 ■■ 56 ■ ■■ ■ ■ * > of the water out 2 . , , 2 ■ 52 ■ ■ ■ 1 of the wheel. 4 3 — 49 — 1+7 1 Received the wa- 5 ■ 4 — — 47 ■ 188 J cer more quietly. 6 5 45 — 225 7 6 42 i 255 8 — — 7 — — 41 — 287 9 ,. 8 38* 308 10 9 36! 328* 11 10 35 i 355 12 — 11 — — 32 £ — — 360^ 13 12 31 i 375 - 14 13 28| 370 j- 15 14 27 i 385 16 15 26 ■ 390 17 16 24 1 392 18 — — 17 224 — — 3864 19 18 -— 21 4 391 1 20 19 20 | 394 }1 Maximum. 21 — 20 19 i 395 J 22 21 184 388i 23 ■ 22 ■ 18 ■■ 396 Work'd irregular. 24 23 — - - Oirerfet by its load. * The fmall difference, in the value of 12 ftrokes of the pump, from the former experiments, was owing to a fmall difference in the length of the ftroke, occafioned by the warping of the wood. Reduction [ 5» ) Reduction of the preceding Specimen* In thefe experiments the head being 6 inches, and the height of the wheel 24 inches, the whole de- fcent will be 30 inches : the expence of water was 14 i ftrokes of the pump in a minute, whereof 12 contained 80 lb.; therefore the water expended in a minute was 967 lb. which, multiplied by 30 inches, gives the power = 2900. If we take the 2 oth experiment for the maximum, we fhall have 20 f turns in a minute, each of which raifed the weight 4! inches, that is, 93,37 inches in a minute. The weight in the fcale was 19 lb, the weight of the fcale io-r oz. ; the counter-weight 3 oz. in the fcale, which, with the weight of the fcale 10 £ oz. makes in the whole ao-lb. which is the whole refiftance or load : this, multiplied by 93,37 inches, makes 1914 for the efFedt. The ratio therefore of the power and effeB will be as 2900 : 1914, or as 10 : 6,6, or as 3 : 2 nearly. But if we compute the power from the height <5f the wheel only, we fhall have 96 •§ lb. multiplied by 24 inches = 2320 for the power > and this will be to the effeSi as 2320 ; 19*4, or as 10 : 82, or a* 5 : 4 nearly. . The reduction of this fpecimen is fet down in N°. 9. of the following Table $ and the reft were die- dufted from a fimilar fett of experiments, reduced ia the fame manner. Table r r 1 3* 3 Table IL containing the Refujt of Sixteen Setts of Experiments on- Over/hot Wheels. ■•** ■* 2 1 2 3] 4 1 o , 6 7 8 . 9 10 12 :s Inch, 2.7 27 27 *7 284 z8i S ■a 4-» AS ca a 3 c fi B s B h'R 3° 3.0 30 56!- 6 |f 76} 7|t 96! "Hi S 3 " s is -2 90 9 6f "3t 19 20 j 21* 6f 18$ 20J 35 33 33 3S 35 '• 2 - 56f io6f 146! 65 I20 163^ to 20 J 21 id 5 ** ~ g £ O •1 53014360 '7i 20j *9l 20i 2 3i 810 1530 «• is .. 04 u J3 & :i 720 I360 I71O 207O 2O9O *7SS 2700 2960 3400 1840 1764 2320 2*160 .2? 20 2720 556 .1060 1167 1500 22; I -9l 21 i *5 211 27! 1 61 z6\ 1870 3520 4840 2275 4200 }7& 6. 1360 2560 3520 1476 <868 7SS 1914 222 10: 6,9 iq : 6,9 1 o : 7, 6 10 : 7,3 10 : 7,3 u o Jq - .2 "tj (2 °- 1 10:7, .10 : 6,8 10 : 6,$ 10: 6,6 10:6,5 10:7,7 10 : 7fi jo :6,4 10 : 8,2 10: 8,2 o H 00 10 : 8,4 10 : 8, 1560 2880 89*4 1230 2153 2846 1466 2467 2.981 8. 10:6,6 10 : 6,1 10 : 5,9 10 :8,i 10: 8,2 10-: 8,2 S 9. 00 00 ? 10 : 6,5 10:5,9 10; 5,2 10:9, to : 8,4 ro : 8,1 o io:9»4 10: $ 9 6 10: 7,6 9. I 10. do- ll *• Obfervations and r £)edu£iions from the foregoing Ex* periments. I. Concerning the Ratio between the Power and Effift of Over/hot Wheels. The effedtive power of the water muft be rec- koned upon the whole defcent j becanfe it muft be 4 raifed j *aifcd tfett height* in Order to be in a condition of producing the ftme effect a fecond time. The ratio's between the powers fo eftimated, and the effe&s at the maximum deduced from the feveral ftm of experiments, we exhibited at one view in column 9. of Table H. ; and from hence it appears, that thafe ratio's differ from that of 10 to 7,6 to that of k>: ft*, that is, nearly from 4: 3 to 4: 2. In thofe {experiments where the heads of water and quantities expended are foaft* the proportion kneasv ly 4s 4 : 3 j but where the heads aad quantities are greateft, it approaches nearer to that of 4 : 2 ; and by a medium of the whole, the ratio is that of 3 : z nearly. We have feen before* in our obfervations upon the effects of underfoot wheels, that the gene- rail ratio of the power to the effe£, when greatefl* was 3:1; the effeSl therefore tfoverjhot wheels \ under the Jhme circumjtances of quantity and fall, is at a medium double to that of the underfoot : and, as a confequence thereof, that nonelaftic bodies^ when a£l- ing by their impuffe or coffijkn, fohtmtmicate oniy a fart of their original power \ the other part beifcg ibent in changing their figure in confequence of the ltroke. The powers of water computed from the height of the wheel oniy, compared with the effefts, as in cohirnn 1 o. appear to obfcrve a more conftant ratio : for if we take die medium of each clafs, which is fef down in colnmn 1 1 % we fliall find the extremes to dfflfer no more than from the ratio of 10 : 8,1 to that tif to : 8,5 5 and avthe fecond term of the ratio «*r&d&atty increafes fromS,t toS,f, by an increale rf head from 3 inches to ri, tfae excefs of 8, J above E 5,i r i [ 34 I 8,1 is to be imputed to the fuperior impulfe of the water at the head of n inches above that of £ inches : fo that if we reduce 8,i to 8, on account of the impulfe of the 3 inch head, we jhall have the ratio of the power, computed upon the height of the wheel only, to the effeSi at a maximum as 10 :S, or as 5 : 4 nearly : and from the equality of the ratio between power and efFedt, fubfifting where the con- ftrudtions are fimilar, we muft infer, that the effefts, as well as the powers, are as the quantities of water and perpendicular heights multiplied together rejfye&ively \ < II. Concerning the mofl proper Height of the Wheel in proportion to the whole Defcent. We have already feen, from the preceding ob~ fervation, that the eflfecft of the fame quantity of water, defcending thro' the fame perpendicular fpace, is double, when adting by its gravity upon an over^ .(hot wheel, to what the fame produces when adting by its impulfe upon an underfhot. . It alfo appears, that by increafing the head from 3 inches to 1 1, that is, the whole defcent, from 27 inches to 35, or in the ratio of 7 to 9 nearly, the effedt is advanced no more than in the ratio of 8,1 to 8,4, that is > as 7 : 7,26; and confequently the increafe of efFedt as not i-7th of the increafe of perpendicidar height Hence it follows, that the higher the wheel is in proportion to the whole defcent 7 the greater will be the effeft ; be*- caufe it depends lefs upon the impulfe of the head, and more upon the gravity of the water in the buckets: and if we confider how obliquely the water iffuing from the head muft ftrike the buckets^ we (hall not be at a loft to account for the little ad- vantage [35] vantage that arifes from the itapulfe thereof ; and fhali immediately fee of how little cohfequence this impulfe ds to the efle£t of an dverfhot wheel. However, as .every thing has its limits, fo has this : for thus much is xlefirable, that the water Jhould have fomewhat greater velocity > than the circumference of the wheel, in coming thereon •, otherwife the wheel will not only <be retarded, by the buckets ftriking the 1 water; but thereby dafliing a part of it over, fo much of the power is loft. The velocity that the circumference of the wheel ought to have, being known by the following de- du&ions, the head requifite to give the water its pro- per velocity is eafily computed from the common rules of hydroftatics $ and will be found much lefs than what is generally prattifed. i< ■ ~ III. Concerning the Velocity of the Circumference of the Wheel \ in order to produce the greateft EffeB. If a body is let fall freely from the furface of the head to the bottom of the defcent, it will take a certain time in falling ; and in this cafe the whole a&ion of grayity is fpent in giving the body a certain velocity : but if this body in falling is made to aft upon fome other body, fo as to produce a mechani- cal effedt, the falling body will be retarded ; becaufe ;apart of the adtion of gravity is then fpent in pro- ducing the effedt, . and' the remainder . only giving motion tQ ihe falling body : and therefore toe flower ja body deft ends, the greater will be the portion of the aftion of gravity applicable . to , the producing a me- chanical jffeft \ and in confequence the greater that "pfFeft may be; ^ ^ E 2 U If a flrtam of water falU ioto tfte bucket of an .over&ot wheel, it is there retained till the wheel by moviog round difeharges it : of confequeace the flower the wheel moves, the more water each bucket •will receive : fo that what is loft m fpeed* is gained by the preflure of a greater quantity of Water a&tng in the buckets at once : and, if confidered ody in this light, the mechanical power of m ovcrfhot wheel to jjroduce effects will be equal, whether it moves quick or flow : but if we attend to what has been juft now obfervod of the felling body, it wifl appear that fo much of the ax^an of gravity, as j» empfoyai ia giving the wheel and water iberon a greater velocity* couft bef ubtraded from its pcefiWre upon the buckets 5 fo that, ttho' the prated: made by mukiplying the number of cubic inches tof water a&iisjg; in the what at once by its velocity will be the fame in all cafes * yet, as each eubfc inch, when the metecky is greater does not pre& fo much upon the bucket as when it is left, the power of the water to produce effects will be greater in the lefs velocity than in the greater z and hence we are led to tnis general rule, that r caeteris paribus, the lefs the velocity rf. the wheel, the* greater will be the <effe& therepf. A confirmation of this doftrine, together with the limits it is fubjeft to in praftice, may be deduced from the foregoing fpe- cimen of a fett of experiments. Fromthefe experiments it.appears,. that when the: wheel made about 20 turns in a minute, the effedt: was, near upon, the greateft. When it made 30 turns, the effect was diminifhed about -jV part $ but that when it made 40, it was dimimifhed about £; when* it made left than 18 i, its motion was irregular ; and when* r 37 j • when ft was loaded fo as not to admit its making i $ turns, the wheel was overpowered by its load. It is an advantage in praftice, that the velocity of the wheel fhould not be diminifhed further than what will procure fome folid advantage in point of power ;. becaufe, catfris paribus, as the motion is flower* the buckets ipuft be made Urger; and the wheel being more loaded with water, the ftrefs upon ever^ part of the work will be increafed in proportion : The befi velocity for pra&tice therefore will befucb y at when the wheel here ufed made about 30 turns in a minute \ that is, when the velocity of the circum- ference is a little more than 3 feet in a fecond. Experience confirms, that this velocity of 3 feet in a fecond is applicable to the higheft overfhot wheels,, srs well as the loweft 5 and all other parts of the work being properly adapted thereto, will produce very nearly the greatefjfc *f&£t poffihle : however this alfo is certain from expedience, that high wheels may deviate further from this rule, before they mil lope their poiper r by a given aliquot part of the whole y than low ones can he admitted to do y for a wheel of 64 feet high may move at the rate of fix feet per fecond without lofipg any conlktacablc part of its fiower * y and, on toe other hand, I have feen a wheel of 33 feet high, that has moved very fteadily and well with a velocity but little exceeding a feet*. * The 24 feet wheel gping at 6 feet in a fecond feems owing to Jbe final! proportion that the head (ueqiiifite to give the water the f^oper velocity #f jfe whsel) bean to d* whole height. IVY Con- • • * IV. Concerning the Load for ; an, Over/hot Wheel^ in ' ,: ; order, that it may produce a Maximum. \ The maximum Joad for an over/hot wheel, is that -which reduces the circumferences of the wheel to its proper velocity - y and this will be known, by dividing ^the effe£t it ought to produce in a given time by the fpace intended to be'defcribed by the circumferefrce of the wheel in the fame time : the quotient will be the refiftance overcome at the circumference of the wheel ; and is equal to the load required, the fric- tion and refiftance of the machinery included. V. Concerning the great eft pojfible Velocity of an . Over -/hot Wheel. :, The greateft velocity that the circumference of an <overftiot wheel is capable of, depends jointly upon *he diameter or height of the wheel, arid the velo- city of falling bodies $ for it is plain that the velocity of the circumference can never be greater, than to defcribe a femi-pircumference, while a body let fall from the top of the wheel will defcend thro' its di- ameter^ nor indeed quite fo great, as a body de- scending thro' the fame perpendicular fpace cannot perform the <fame in fo fmall a time when pafiihg thro* a femi-cirole, as would be done in a perpendi- cular line. Thus, if . a wheel is 1 6 feet i inch high, a~body will fall thro' the diameter in one fecond: jthis whed therefore can never af rive at a velocity equal to the making one turn in two feconds ; but, in reality, an overftiot wheel can never. come near this velocity; for when it acquires a certain fpeed, j the E 39 3 ttie greateft part of the water is prevented" from en- tering the buckets $ and the reft, at a certain point of its defcent, is thrown out again by the centrifugal force. This appears to have been the cafe in the three firftf experiments of the foregoing fpecimen ; but as the velocity, when this begins to happen, de- pends upon the form of the buckets, as well as other circumftances, tbe^utmoft velocity of over/hot wheel* is not to be determined generally : and, indeed, it is the lefs neceffary in pra&ice, as it is in this circum- ftance incapable of producing any mechanical effect ^ for reafons already given. VI. Concerning the greatejl Load that an Over/hot Wheel can overcome. m The greatejl load an over {hot wheel will overcome ^ eonfidered abfiradiedly, is unlimited or infinite : for as the buckets may be of ahy given capacity, the more the wheel is loaded, the flower it turns / but the flower it turns, the more will the buckets be filled with water ; and confequently tho' the diameter of the wheel, and quantity of water expended, are both limited, yet no refiftance can be afligned, which it is not able to- overcome : but in practice we always meet with fomething that prevents our getting inro infinitefimals 5 for when we really go to work to build a wheel, the buckets muft neceflarily be of fomc given capacity; and confequently fuch a rejijlance willjlop the wheels as is equal to the effort of all the buckets in one femi-circumference filled with wafer. The ftru&ure of the buckets being given, the quantity of this effort may be afligned $ but is not of much confequence to the practice, as in this cafe alio [ 4*> ) alfo the vrheel lofts its power j for tho 9 herd is thft exertion of gravity upon a given quantity of water, yet being prevented by a counterbalance frocq mov- ing, is capable of producing no mechanical ejfie&i according to our definition. But, in reality* an over-* {hot wheel ^generally ceafes to be ufeful before it if loaded to that pitch; for when it mtets witbjkcha rtfftance as to diminijh its velocity H a certain degree^ its motion becomes irregular $ yet this never happens till the velocity of the circumference is lefs than zjeet ferfecond > inhere the rejiftanct *j tquakle, as appear* not only from the preceding fpecimen, but from ex- periments on larger wheels. Scholium* Havkvg now examined the different effeds of the power of water, when acting by its impulfe^ and by its weighty under the titles of underjhot and overjbtt wheels ; we might naturally proceed to examine the effe&s when the imputfe and weight are combined, as in the feveral kinds of br*aft-wbeels % &c. but, what has been already delivered being carefully attended to, the application of the fame principles in thefe mixt cafes will be eafy, and reduce what I have to &y on this head into a narrow compafs : for all kinds of wheels where the water cannot descend thro' a given fpace § imkfs the wheel moves therewith, are to be con* fidered of the natttre of an overihot wheel, accord- ing to the perpendicular height that the water de~ fcends froto ; and all thofe that receive the impuMb 4&r (hock of die water > whether in an horizontal* per- pendicular, or oblique direction* Are to be considered B& understate Atd thetfefer e a wk*l> which the water [4i] water ftrikes at a certain point below the furface of the head, and after that defcends in the arch of a circle, preffing by its gravity upon the wheel ; the effeB offuch a wheel will be equal to the effeSl of an under/hot, whofe bead is equal to the difference of level between the furface oj the water in the refervoir and the point where itjlrikes the wheel, added to that of an over/hot, whofe height is equal to the difference of kvel> between the point where it Jlrikes the wheel and the level of the tail-water. It is here fuppofed, that the wheel receives the (hock of the water at right angles to its radii ; and that the velocity of its circumference is properly adapted to receive the ut- moft advantage of both thefe powers ; otherwife a reduction muft be made on that account. Many obvious and considerable improvements up- on the common pradtice naturally offer themfelves, from a due confideration of the principles here eftab- lifhed, as well as many popular errors fhow them- felves in view : but as my prefent purpofe extends no farther than the laying down fuch general rules as will be found to* anfwer in pra&ice, I leave the particular application to the intelligent artift, and to- the curious in. thefe matters, m P A R T lit On the ConJlruSlion and EffeSts c/WrNDMkL- Sails. Read 31 May &r | N trying experiments on windmill- 14 June, 1759- 1 f ai i s> ^ wind itfelf is too uncertain to anfwer the purpofe : we muft therefore have.re- courfe to an artificial wind. F Tbi* N "5 [4»1 - This. : njay [bp/4<Hie two. ways.; either by^ca^fiog the air to moye againft tlj? machine, or the raa$hif*r to move again^ the air. * To canfe the air to ih&ve ^ainft the machine, ia a fufficiefit vplumn, with fteadinefs and. the requifit? velocity, is n$t eafily put in pra&ice: To parry the machine forward in a jright line again ft the air, wquH require: a larger room than I could conveniently meet with. What I found moft practicable, therefore, was,, to carry the axis, ^whereon the fails were to b^fixsd, progreflivfely round. in the circumference of 3 large circle. Upon this idea * a machine wa$. conftrufted, . as follows* Plate VI. Fig. i. ABC is a pyramidical frame for - fuppprting the moving parts. D E is an upright axis, f whereon is framed F G, an arm for carrying the fails- at a proper dis- tance from the center of the upright axis. ■ ■ ■ ■ i > * Some years ago Mr. Roufe, an ingenious gentleman of Har- borovgh in Letcefterikire, fet about trying experiments on the ve- locity of the wind, and force thereof upon plain, fucfaces and winarnill-faUs : and much about the fame time Mr, EUicott con- trived a machine for the ufe of the late* celebrated Mr. B, Robins, for trying the refiftance of plain furfaces moving thro' the air.- The machines of both thefe gentlemen were much alike, tho' ^t that time totally unacquainted with ^ach other's inquiries. But it often happens, that wheii two perfons think juftly upon th$ fame iubjed, their experiments are aJjke. This machine was alfo.built, upanJhe-fime idea is th$ foregoing ; but dirrered in having the hand for the firft mover, with a pendulum for its regulator, inftead of a weight, as in the former 5 which was certainly beft for the pur- ipofes ofmeafuring the impdfe of the windy or refiftance, b£ plain* : -but the latter is more applicable : to. experiments on windmill-fail* \ ^becaufe every change^ of pofttion of tr*£\"ame fails will occafion their meeting the>ak wU^a4iffi^ent- Vekwaty, -tho- urged by the- jfejne weight. [43 J. If is a barrel Upon the upright axis, ~ whereon is wound a cord ; which* being drawn by the hand, gives a circular motion to the axis, and* to the arm F 6 ; and thereby carries the axis of the fails in the circumference of a circle, whofe radius is D.I, caufing thereby the fails : to ftrike the air, and turn round upon their own axis. At L is fixed the end of a fmall line, which paffing through the pullies M"N O, terminates upon a fmall cylinder or barrel upon the axis of the fails j and, by winding thereon, raifes* F the fcale, wherein the weights are placed for trying the power of the fails* This fcale, moving up and down in the direction of the upright axis, receives no^diflurbance from the circular motion. <^R two parallel pillars iknding upon the arm FG, for the purpofe of fupporting and keeping fteady the fcale P$ which is kept from fwing- ing by means pf S T two fmall chains, which hang loofely round the, . two pillars. W is a weight, for bringing the center of gravity of the moveable part of the machine into the cen- ter of motion of the axis D E. VX is a pendulum* compofed of twp balls of lead,, which are moveable upon a wooden rod, and* thereby can be fo adjufted, as to vibrate in any time required. Thiy pe nd u lu m hangs upoft a : cylindrical wire, whereon it vibrates, as on a: rolling axis. Y is a perforated table for fupporting the axis of the pendulum, E z~ Nvte„ * \* [ 44 ] Note, The pendulum being fa adjufted, as to make two vibrations in the lime that the arm F G is in- tended to make one turn ; the pendulum being fet a vibrating, the experimenter pulls by the cord Z, with fufficient force to make each half revolution of the arm to correfpond with each vibration, as equal ,as poffible, during the number of vibrations that the experiment is intended to be continued. A little practice renders it eafy to give motion thereto with all the regularity that is neceflary. Specimen of a Sett of Experiments. Radius of the fails — - — 2 1 inches Length of ditto in the cloth — — 18 Breadth of ditto — 5,6 % $ Angle at the extremity — — *— 10 degrees {, Ditto at the greateft inclination — 25 20 turns of the fails raifed the weight 11, 3 inches Velocity of the center of the fails, in the! circumference of the great circle, in a>6f l . o in. fecond ■— — - — — — J Continuance of the experiment 52 feconds. N°. Wt. in the fcale. Turns. Produ<». I olb. 108 O 2 — ^ 6 85 — - . 510 3 6f 8* — fi6i ^ _- y — y% -*, — £^(y $ 7*— 73 j+7f maxim* 6 — r-* S *6f 520 <F-* * In all the following experiments the angle of the fails is ac- counted from the plain of their motion ; that is, when they ftand at right angles to the axis, their angle is denoted o°, this notation being agreeable to the language of pra&itioners, >vho call the angle fo denoted, the weather of the fail j which they denominate greater or lefs, according to the quantity of this angle. N.B* * [45] N.B. The weight of the fcale and pulley was 3 pz.; and that 1 oz. fufpended upon one of the radii, at 12 1 iches from the center of the axis, juft over- came the fridtion fcale and load of 7 i lb. 5 and placed at 14 £■§■ inches, overcame the fame refin- ances with <? lb. in the fcale. Reduction of the preceding Specimen. N°. 5. being taken for the maximum, the weight in the fcale was 7 lb. 8 oz. which, with the weight of the fcale and pulley 3 oz. makes 7 lb. 11 oz. equal to 1 23 oz.; this added to the friftion of the machinery, the fum is the whole: refiftance *. The fri&ion of the machinery is thus deduced : Since 20 turns of the fails raifed the weight 1 1,3 inches, with a double line, the radius of the cylinder will be .18 of an inch -, but had the weight been raifed by a fingle line, the radius of the cylinder being half the former, viz. .09, the refiftance would have been the fame: we (hall therefore have this analogy; as half the radius of the cylinder, is to the length of the arm where the fmall weight was applied ; fo is the weight applied to the arm, to a fourth weight, which is equivalent to the fum of the whole refiftance to- gether; that is, .69 : 12,5 : : i oz. : 1390Z.: this exceeds 123 oz. the weight in the fcale, by 1 6 oz. or 1 lb. which is equivalent to the fridtion ^ and which, added to the above weight of 7 lb. 11 oz. makes $ lb. 11 oz. = 8,6p lb. for the fum of die whole re- * The refiftance of the air is not taken into the account of refiftance, "bbcaufe it is infeparable from the application of the power. 4 iiftance* T46J fiffancc; and this,. multiplied by 7 j turns, makas * product of <Jj4, which may be called the representa- tive of the effeSt produced. In like manner, if the weight 9 lb. which caufpd the (ails to reft after being in motion, be augmented by die weight of the fcale and its relative fri&ion, it will become 10,371b. The refult of this fpecimen, is fet down in N°. 12. of Table III, and the refult of every other fett of experiments therein contained, were made and reduced in the fame manner. Tabljt *& •* - C 47 3 Table HI, Containing Nineteen Setts of Experiments onWindmilUSails of various Structures, Portions, and Quantities of Surfaces. The kind of fails made ufe of. o' Plain fails it att angle of 55°. { P/*r>r fails weatherM C according to the< common pra&ice. C Weathered aceord-C ing to Macl4urin > s\ theorem. L Sails weathered in the Dutch man- ner, tried in va- rious pofitions* Sails weathered 1 in the Dutch manner, but enlarged to- wards the extremi- ties. 8 fails being fefio'rs f of elliffes in-theirx- beft portions. ( t r 1 Pi I 8 I 7 8 9 10 11 12 *3 H *5 16 «7 18 1? 1. o 35 1* 18 J 9 12 -.3 5 7i to 12 7* 10 12 »5 12 12 "2. f 1 o 35 T2 '5 18 *6{ 32i >5 20 * 2 f 25 27 2 «: 25 27 30 22 22 3- 4S 66 105 96 tZOl 120 108 IOO 123 "7 114 96 99 O 6* O g s «* r . *-» IT* cd 42 70 69 66 f o- 7*5 6 6& 63 93 79 7* 77 7* 66 7S 74 66 63 642 6 4 i 5- 6,72 7>o 7>° 8.3 4*75 7»o 7»5 8,3 ft*9 8,41 10,65 11,08 1 2,09 12,09 Hi s 12,59 7»5 6 8,12 9*8* *fM2 18,06 6. 5>3« 8»I2 8,12 9.81 »<*»37 10,94 12,59 13,69 ■4>*3 J 4^7 8 27,87 »«M> JL 4 1 Oh i. 318 <*4 o a C/ <2 •* 5* B JJ > B 8.8 1 •St 404 10:7 441 464 462 ,404 f 404 404 462 51^ 527 442 553 585 639 634 •580 799 820 799 762 J°59 1165 8. 404 404 4«>4l 404 4«>4 404 404 4*>4 404 5°5 5°5 5<*5l 5°5 .854 1146 •£ io: 6,6 10:7, 10:7,7 10:6,6 10:6,8 10:6,8 10:6,6 10:6,1 10:6,3 to 15,8 10:6,6 io:6,]i 10:5,9 10. . o ** a " *« 2 c<-2 £ 10:6 10:8,3 10:8,3 10:7,1 10:8,5 ro:8,i ib : 8,4 io;8,2 «l-5»9 11. £4 o ^ •S-3 10:7,9 ib: 8,9 ftp: 8,6 10:9,2 10:8,5 J& : 8,4 W):7,7* 10: 10,1 \o: 10,15 10:10,15 10: 11,4 10: 12,8 10:13, IO*II, 0:13,7 10:14,5 10:15,8 10:15,7 10:14,4 10:15,8 10:16,2 10:15,8 10: 15,1 10:12,4 IO:iO,I 12. ', . > hX ^tytfervatfon* [4»] Obfervations and Deductions from, the preceding 'Experiments. I. Concerning the bejl Form and Pofition of Wind- mill-Sails. In Table III. N°. i. is contained the refult of a fett of experiments upon fails fet at the angle which, the celebrated Monf. Parint, and fucceeding geome- tricians for many years, held to be the beft ; viz. thofe whofe planes make an angle ff° nearly with, the axis ; the complement whereof* or angle that the- plane of the fail makes with the plane of their mo tion, will therefore be 35 , as fet down in col. 2. and 3. Now if we multiply their number of turns by the weight they lifted, when working to the greateft advantage, as fet down in columns 5. and 6. and* compare this product (col. 8.) with the other pro- ducts contained in the fame column, inftead of being the greateft, it turns out the leaft of all the reft. But if we fet the angle of the fame planes at fome- what lefs than half the former, or at any angle from* 1 5® to 1 8°, as in N°. 3. and 4. that is, from 72 to 7f° with the axis, the produdt will be increafed in. the ratio of 3 1 : 45* -, and this is the angle moft com*- monly made ufe of by practitioners, when the fun- faces of the fails are planes. If nothing more was intended than to determine the moft efficacious angle ta make a mill acquire motion from a ftate of reft, or to prevent it from pafling in to- reft from a ftate of motion, we fhall find the pofition of N°. 1. the beft ; for if we confult col. 7. which contains the leaft weights, that would make the fails pafs from motion tQ re A, we fhall find that of N°. 1. (relative, [ 49 ] (relative to the quantity of cloth) the greateft of all. But if the fails are intended, with given dimenfions, to produce the greateft effedt poftible in a given time, we muft intirely reject thofe of N°. i. and, if we are confined to the ufe of planes , conform ourfehes to Jome angle between N°. 3. and 4. that is, n$t lefs than 7 **> or greater than 7 j> °, with the axis. The late celebrated Mr, Maclaurin has judicioufly distinguished between the action of the wind upon a fail at reft, and a fail in motion ; and, in confequence* as the motion is more rapid near the extremities than towards the center, that the angle of the different parts of the fail, as they recede from the center, ihould be Varied. For this purpofe he has furnifhed us with the following theorem *. c< Suppofe the velocity <f of the wind to be reprefented by a, and the velo- a city of any given part of the fail to be denoted by €i € ; then the effort of the wind upon that part of the fail will be greateft when the tangent of the an gle, in wh ich the wind ftrikes it, is to radius as « 1 2, 4. .iff _{- If to 1." This theorem then af- figns the law, by, which the angle is to be varied ac- cording to the velocity of each part of the fail to the wind : but as it is left undetermined what velocity any one given part of the fail ought to have in rcfpc^i to the wind, the angle that any one part of the fail ought to have, is left undetermined alfo ; fo that we are ftill at a lofs for the proper data to apply the theo- rem. However j being willing to avail myfelf thereof, and confidering that any angle from 1 5 to 1 8° was beft fuited to a plane, and of confequence the beft * Maclaurin's account of Sir Ifaac Newton's philofophical dif- covcries, p. 176, art. 29. G mean [5P] mean angle, I made the fail, at the middle diftancr between the center and the extremity, to ftand at an. angle of 1 5 41' with the plane of the motion 5 in which cafe the velocity of that part of the fail, whea loaded to a maximum, would be equal to that of the wind, or c » a. This being determined, the reft were inclined according to the theorem, as follows : Angle with Angle of the axis. weather. f£ - - c — \a - - 6f 26'-- 26* 34' Parts of theU--' = i*-- 6 9 54 - - 20 * radius from< f - - € =Z a - - 74 19 - - 15 41 middfe the center. * - - f = j|* - 77 20 - - 12 40 . i--* = aa--8i o - - 9 o extremity.. The reful t hereof was according to ft°. f. being nearly the fame as the plane fails, in their beft por- tion : but being turned round in their fockets, fo that every part of each fail flood at an angle of 3 , and afterwards of 6% greater than before, that is, their extremities being- moved from 9 to 12* and if , the products were advanced to ji$ and 527 refpedtively* Now from the fmall difference between thofe two> produ&s, we may conclude, that they Were nearly in their beft pofition, according to N°. 7. or fome angle between that and N 9 . 6 : but from thefe, as well as the plane fails and others, we may alfo conclude* that a variation in the angle of a degree or two. makes very little difference in the. effeSi y when the angle is near upon the bejl. It is to be obferved > that, a foil inclined by the preceding rule will expofe a convex furface to the wind : whereas the Dutch* and all our modern xniU- t 5* ] mill-builder*, tho^they make the angle to diminifiv in receding from the center towards the extremity, yet constantly do it in fuch manner, as that the far- face of the fail may be concave towards the wind. In this manner the fails made ufe of in N°. 8, 9/10, ii, 12, and 1 3. were conftru&ed ; the middle of the fail making an angle with the extreme bar of 1 z\ and the greateft angle (which was about -£• of the ra- dius from the centre) of 1 j° therewith, Thofe fails being tried in various portions, the beft appears to be that of N°. 11* where the extremities ftood at an angle of 7°! with the plane of motion, the produdfc being 639 : greater than that of thofe made by the theorem in the ratio of 9 : 1 1, and double to that of N°. i.j and this was the greateft product that could be procured without an augmentation of furface. Hence it appears, that when the wind falls upon a concave fur j ace > it is an advantage to the power of the whole 7 tho % every party taken feparately^Jhould not be difpofed to the beft advantage *. Having thus obtained the beft pofition of the fails, or manner of weathering, as it is called by workmen,; the next point was to try what advantage could be MMMHMMBMMMHMWMM*««a>-MHMMMiii * By feveral trials in large I have found the following angles to anfwer as well as any. The radius is fuppofed to be divided into 6 parts and i-6th, reckoning from the center, is called 1, the ex- tremity being denoted 6. Angle with Angle with the plane N°. the axis* of motion. 1 72° — 1 8* 2 71 ■ 19 2 -, , 72 - - 18 middle 4 , , 74 „.„ 16 5 77i ■ ■ ia| 6 ■■ 83 ■ ■ • ■ 7 extremity. G 2 made [5* J made by an addition of furface upon the fame ra- dius* For this purpofe, the fails made ufe of had the fame weather as thofe N°. 8. to 13, with an addition to the leading fide of each of a triangular cloth, whofe height was equal to the height of the fail, and whofe bafe was equal to half the breadth : of confequence the increafe of furface upon the whole was one fourth part, or as 4 : f. Thofe fails, by being turned round in their fockets, were tried in four different pofitions, fpecified in N°. 14, 15, 16, and 17; from whence it appears, that the beft was when every part of the fail made a greater angle by 2°|, with the plane of the motion, than thofe with- out the addition, as appears by N°. 15. the product being 820 : this exceeds 639 more than in the ratio of 4 : 5, or that of the increafe of cloth. Hence it appears, that a broader fail requires a greater angle ; and that when the fail is broader at the ex- tremity v than near the center *, thisjhape is more ad- vantageous than that of a parallelogram *. f Many have imagined, that the more fail, the greater the advantage, and have therefore propofed to fill up the whole area : and by making each fail a fedtor of an ellipfis, according to Monfieur Paring to intercept the whole cylinder of wind, and thereby to produce the greateft effed poflible. «■■■ I' I ' ' ' I I ■ ■ f II ■ H I II I ■ * The figure and proportion of the enlarged fails, which I have found beft to anfwer in large, are reprefented in the figure, Plate VI. where the extreme bar is i-ld of the radius (or whip, as ft is called by the workmen), and is divided by the whip in the proportioa *>f 3 t° 5* The triangular or leading fail is covered' with board from the point downwards i-3d of its height, the reft with cloth as ufual. The angles of weather in the preceding note are beft for the enlarged fails aHb ; for in pra&ice it is found, that the fails |t£d better have too little than too much weather. X [53] We have therefore proceeded to inquire, how far the effect could be increafed by a further enlargement of the furface, upon the fame radius of which N Q . 1 8 and 1 9 are fpecimens. The furfaces indeed were not made planes, and fet at an angle of 3 $°, as Parint propofed ; becaufe, from N°. 1. we learn, that this pofition has nothing to do, when we intend them to work to the greateft advantage. We therefore gave them fuch an angle as the preceding experiments in- dicated for fuch fort of fails, viz. 1 1° at the ex- tremity, and 22 for the greateft weather. By N°. 18 we have the product 1079, greater than N°. 15. in the ratio of 7 : 9 3 but then the augmentation of cloth is almoft 7:12. By N°. 19. we have the pro- duct ii6f y that is greater than N p . if. as 7 : 105 but the augmentation of cloth is nearly as 7 : 165 confequently had the fame quantity of cloth -as in 1^°. 18. been difpofed in a figure fimilar to that of N°. 15, inftead of the produtt 1059, we fhould have had the produdt 13865 and in N°. 19, inftead of the produ<ft 1 1 6f, we fhould have had a produd: of i860; as will be further made appear in the courfe of the following deductions. Hence it ap- pears, that beyond a certain degree, the more the area is crowded with fail, the lefs effect is produced in proportion to the furface : and by purfuing the experiments ftill further, I found, that tho' in N°. 19. the furface of all the fails together were not more than 7~8ths of the circular area containing them, yet a further addition rather diminifhed than increafed the effeft. So that when the whole cylinder cf wind is intercepted, it does not then produce the greateft effeftfor want of proper interftices to efcape. [ 54 ] It is certainly defirable, that the (ails of windmills fliould be as lhort as poffible ; but at the fame time it is equally defirable, that the quantity of cloth fhould be the lead that may be, to avoid damage by fudden fqualls of wind. The beft ftru&ure, there- fore, for large mills, is that where the quantity of cloth is the greateft, in a given circle, that can be ; on this condition, that the efFed holds out in pro- portion to the quantity of cloth ; for otherwife the effect can be augmented in a given degree by a lefler increafe of cloth upon a larger radius, than would be required, if the cloth was increafed upon the fame radius. The mod ufeful figure therefore for practice, is that of N°. 9. or 10. as has been experienced upon feveral mills in large. Table [•55] a* ^* 1 Si ** O C § v S ** ; u CM 6& 2 S "I- 3 c j s J S % <3 as ■ Ratio of the greateft load to the load at a maximum. co m * 0* oo 0\ • • • . o o 1 1 00 00 • • »• O Ratio of the grcateft velocity to the ve- locity at a maxim™. OS OS 0k 0h vo vo • • »• O o 1 1 t^ CM 1 vo vo • • •• o o •O M Ratio of the two produces. IS CM : iil IS hi Ivo CM 1 o Produftofleflerload and greater velocity. 1 ° 1 oo | <o 1 00 vo 1 OS 1 tx « Tarns of the fails therewith. o 1*2 \i OO 1? o Maximum load for the half velocity. 15 CM CO 1 "vr> OS Product vo CO Os Q CM O cm O «0 ci> cm CM o ^ co O 00 * Greateft load. • *^VO ^* vooo 1 1 oo co vo *• CM IH Load at the maxi- mum. rx cm Att CM CM VO vo * * CO w o vo * * vooo m VO Turns of the fails at maximum. VO CM VO CM vo O VO co M M Q VO m vo Turns of the (ails unloaded. vo t>s oso * I" ^ oo <+ Velocity of the wind in a fecond. ,4 H* XT' OS ^■OO ^- OS ^oo CO Angle at the extre- mity. vo us. o o 1-4 »N CM N° •1 CM 1 1 co ^t- I VOVO 1 M % II. Con- r s6 3 II. Concerning the ratio between the vebtity of windmill fails unloaded \ and their velocity when loaded to a maximum. Thofe ratio's, as they turned out in experiments upon different kinds of fails, and with different in- clinations (the velocity of the wind being the fame) are contained in column i o of tab. III. where the extremes differ from the ratio of 10 : 7,7 to that of 10 : 5,8 ; but the mojl general ratio of the whole will be nearly as $ : 2. Thus ratio alfo agrees fufficiently near with experiments where the velocity of the wind was different, as in thofe contained in tab. IV. col. 13. in which the ratio's differ from 10 : 6,9 to that of I o : f,5>. However, it appears in general, that where the power is greater, whether by an enlargement of furface, or a greater velocity of the wind, that the fecond term of the ratio is lefs. III. Concerning the ratio between the greatejl load that the fails will bear without flopping^ or what is nearly the fame things between the leaft load that will Jlop the fails, and the load at the maxi- mum. Thofe ratio's for different kinds of fails and in- clinations, are colle&ed in col. 1 1. tab. III. where the extremes differ from the ratio of iq : 6 to that of 10 : 9,2 ; but taking in thofe fetts of experiments only, where the fails refpedtively anfwered beft, the ratio's will be confined between that of 10:8 and of 10:9; and at a medium about 10 : 8,3 or of 61 f. This ratio alfo agrees nearly with thofe in col. 1 4 of tab. IV. However it appears, upon the whole, that ■ in thofe inftances, where the angle of the fails or quan- [57] quantity^ of cloth were greateft, that the fccond term oFthte' ratio was lefs* * IV. Concerning the effe&s of fails, according to the different velocity of the wind. Maxim i. The velocity of windmill fails > whe- ther unloaded^ or loaded Jo as to produce a maximum ^ is nearly as the velocity of the windy their Jbape and poftion being the fame. : This appears by comparing together the refpe&ive numbers of columns 4 and 5, tab. IV. wherein thofe of numbers 2, 4, and 6, ought to be double of num- bers 1, 3, and f: but as the deviation is no- where greater than what may be imputed to the inaccuracy of the experiments themfblves, and hold good exact- ly in numbers 3 and. 4 $ which fetts were deduced from the medium of a number of experiments, care- fully repeated the fame day, and on that account are C*6ft to be depended upon ; we may therefore con- clude the maxim true. Maxim 2. The load at the m&o&mum U nearly y but /bfnewbat lefs than, as the fquUre of the, vekcity of the windy the jhape and pofition of the fails be* ing the fame. - • - ' » : • .* This appears by comparing ' together the number? in c6L6. tab. IV. wherein thofe of/ ikihibers V4i and 6 (as" the velocity is double),' ought to* 1)6 qua- druple of thofe of numbers 1, 3, and 5 ; kiftfead of which they fallfhort, number 2. by y#, nflmber'4 by tV>' and number 6 by -^ part >of' dfe .Whole. The greateft of thofe deviations is not more ■ conii- derable than might be imputed to the unavoidable H errors E 5« J wrore \n fiiftking thq experiments; bpt af ikofe experiments, as well as thofe of (lie grea£e# Iqa^ alk deviate the fame way ; and alfo coincide with fome txperime&ts - communicate to tne by Mr. Roufe upon the refifUnce of planes i I am led to fuppofe a* fmall deviation, whereby the load fails fhort of the fquares of the velocity* and. firicp the experiments N° 3 and 4. are moft to be depended upon, we muft -conclude, that when the velocity is double, the* load falls fhort of its due proportion by 7 r 7 , or, for the lake of a rosmd jamjobcr, by about ^ paf t of tfie whole. " r • Maxim 3d- Tie effe&s of the f&nt faiU at a maxi- mum are nearly^ but /amewbat k/s than* as tkp eukei *f the velocity of the fwittd. It has already been proved,. Maxim, lfi> tftatr the velocity of fails at the maximum > is nearly^ as the ve~- locity of the wind ; f and by Maxim 2d, that the load at the maximum is nearly as the fquace of the fame velocity: if thofe two maximums woufd hold pre* rifely, it would be a< canfequence that the effedr would he in. a< triplicate ratio thereof: bow tbia agrees with experiment will appear by ^Gpiparing together the products in col. *» of tab, 4, wherein, thofe of V& z> 4* sod & .(jhc. velocity of the wind being double), ought to be o<9uple sf fchofc of N<> 1. Z* and 5. inftead of which they fall fhwt,,No 2. by $ W° 4, by T ^> aDC * No <J. by 3- part of the whole. Now, if we rely on N° 3 . and 4. as the turns of the fails are as the velocity of the wind ; and fince the load of the maximum falls fhort of the fquare of the yclocky by about ^part of the whole:, the product made [ $9 ] *ntde bf %be mtfttipUGatkm <$f the tarns Irtto theload> fnuft alfo fall (hort of the triplicate ratio by about ^V part of the whole product. Maxim 4th. The had if the fame fails at the maxi- mum is nearly as the fqua^es, and their tfflSl as the tubes, of their humber of turns in a given time^ This maxim fiVay be efteertied a confequence of the three preceding ; for if die turns of the &Us are as the velocity of die wind, whatever quantities are in any given ratio o£ the velocity of the wind, will be in the fame given ratio of the ttjrns of the fails : and therefore, if the load at the tnaximum is as the fquare, or the efFcdfc as the cube, of the velocity of the wind* wanting -~5 P*t when the veldcity is -double ; the load at the maximum will alijo be -as the iquare, and the efFe& as the cube, of the number of turns of the fails in a given time, wanting in like manner ^ part when the number of turns are double in the £ame time. In the pfefent 4afe, if we com- pare the loads at the Maximum coi. 6. with the fquarts of the ntmafod: of tarns col, f. of N° i and .2. f and 6. or the products of the fame numbers col. 8. with the cubes of the number of tarns col. f. inftead of felling fhort, as N° $ and 4. they exceed thofe ratios : but as the fetts of experiments N° 1 and 1* 5 and & are not to fee efteernfed of squil authority with thofe -rf N° 3 and 4. we touft not f ely upon them farther than to obferve, that in tvntyafing the ySF&fi *$*&* rf forg* fn*chine*i the direSt proportion 'fif-theftyuares aird cubes ref$e8dvety % xbiU hid as near jOS the affefih thabjbfoes can be ofyervedt and there- H 2 fore fore be fufficient for pra&ical eftlmation, without any allowance* Maxim jth. When fails are loaded fo as to produce a maximum at a given velocity, and the velocity of the wind increafes, the load continuing the fame ; iftly, The increafe of effe£t r when the increafe of the velocity of the wind is fmall, will be nearly as the fquares ofthofe velocities: idly, When thevehciiy of the wind is double , the effects 'will be nearly as i o : 2 j% : But> $dly, When the velocities compared, are more than double of that where the given bad produces a maximum, the effeSls increafe nearly in a fmple ratio of the velocity of the wind. It has already been proved, maxim ift and 2& y that when the vfclocity of the wind is increafed, the turns of the fails will increafe in the fame proportion, even when oppofed by a load as the fquare of the ve- locity j fcnd therefore if wanting the oppofition of an increafe of load, as the fquare of the velocity, the turns of the fails will again be increafed in a fimple ratio of the velocity of the wind on that account alfo ; that is, the load continuing the fame, the turns of the fails in a given time will be as the fquare of the ve- locity of the wind ; and the effect, being in this cafe ds the turns of the fails, will be as the fquare of the velocity of the wind alfo ; but this muft be under- flood only of the firft increments of the velocity of the wind ; for, 2dly, As the fails will never acquire above a given velocity in relation to the wind, tho' (he load was diminished to nothing \ when the load continues the fame } [6i ] feme, *the more the velocity of the wind* increafes (tho* the effect will continue to increafe) yet the more it will fall fhort of the fquare of the velocity of the wind ; fa that when the velocity of the wind is double, the increafe of effect, inftead of being as 1 14, according to the fquares, it turns out as 10 : 27^, as thus appears. In tab. 4. coh 5). the loads of N° 2, 4/ and 6. are the fame as the maximum lodds in col. 6. of N° 1, 3, and f. The number of turns of the fails with thofe loads, when the velocity of the wind is double, are fet down in col. 10. and the pro- ducts of their multiplication in col. 11 : thofe being compared with the products of N° 1, 3, and 5. coK 8. furnifli the ratios fet down in col. 12. which at a medium (due regard being had to N° 3. and 4.) will be nearly as 10: 27-y. 3dly. The load continuing the fame, grows more and more ihconfiderable, refpe£t- ing the power of the wind, as it increafes in velocity; fo that the turns of the fails grow nearer and nearer a coincidence with their turns unloaded 3 that is, nearer and nearer to the fimple ratio of the velocity of the wind. " When the velocity of the wind is double^, the turns of the fails, when loaded to a maximum, will be double alfo ; but, unloaded, will be no more than triple, by dedudtlon 2d: and therefore the pro- duct could not have increafed beyond the ratio of 10:30 (inftead of 10: 2 7-) even luppofing the fails not to have been retarded at all by carrying the maxi- mum load for the half velocity. Hence we fee, that when the velocity of the wind exceeds the double of that, where a conftant load produces a maximum,, that the increafe of effect, which follows the increafe of the velocity of the fails, will be nearly as the velor city: * % 3 . _ t«3 tclty of the wind* and ultimately Iff that ratio pr&- rifely. ' Hence alfo We fee that Windmills, fuch sis thfe different fftefciei for raiflrig Water for dfakittg*, Afc. lofe faHich of their fell eflfe<a> Wh&i idtihg Sgaiiift one invariable opposition. W. Concerning the effe&s bf fails tf different magni^ tudes, tbt 'jfiru&Ure and p&fition being fimilar> add the velocity yfthe tvind fbejbne* Maxim 6. In fails of a fimUar figure and portion, the number of turns in a given tittle will be recipro- cally as the radius pr length of the fail. > The extreme bar having the fame inclination fb *he plain of its motion, and to the wind, j its velocity at a maximum will always be in a given ratio to the velocity bf the wind -, arid therefore, whatever be the radius, the abfolute velocity of the extremity of the fail will be the fame : arid this will hold good re- 'fpe&ing any other bar, whofe inclination is the fame, ;at a proportionable diftarice from the center ^ it there- fore follows, that the extremity of all fimilar fails, *with the lame /Wind, will have the fariie abfolute velocity 3 and therefore take a f^ace of time to per- form one revolution in proportion to die radius ; or, which is the fame thing, the number of revolutions in the fame given time, will be reciprocally as the length oftheiaii. Maxim 7. The had at a Maximum that fails of ajltnilar figure and portion ittill frOert&ne, at a givek M fiance from the center of motion ', *toill be afs the cuk wf the radius. 4 Ceo- \ Geometry informs us, that in fimilar figures the ffyrfaces are as the fqqares of their fimilar fides ; of confeqjyence the quantity of cloth will be as the fi^are pf the radius : alfo in fimilar figures and pofi- t£op$, the impulfe of the wind,, upon every fimilar fe&ion of the cloth,, will be in proportion to the fur- face of that fedion; and confequently, the impulfe of the wind upon the whole, will be as the furface of the whole : but as die diftance of every fimilar fee- tion, from the center of motion, will be as the ra-> dius ; die diftance of the center of power of the whole, from the center of motion, win be as the ra- dius alfo ; that is, the lever by which the power a&s r will be as the radius : as therefore the impulfe of the wind, refpeding the quantity of cloth, is as the fquare of the radiu$, and the lever, by which it adts,. as the radius fimply ; it follows,, that the load which* the fails will overcome, at a given diftance from the: center, will be as the cube of the radius. / Mmmfy.tfhrejfcdi qf fails of fimilar figure and $oJition y are as the fquare of the radius. m By maxim 6. it is proved, that the number of re- volutions made in a givea time, are as the radius in- vecfely. Under maxim ji. it appears, that the length pf the lever, by which the power a<5ts, is as the radius diredlly ; therefore th$fe equal and oppofite ratios de- ftray oue another : but as in fimilar figures the quan- tity of cjoth is as the fquare of the radius, and the a<$tion of the wind is in propbrtipn to the quantity of sl^th, as alfo appears under qiaxim 7 ; it follows that the eftefit is as the fquare of the radius. Gorow « -.. [6 4 ] X^orol. i. Hence it follows, that augmenting Are 3ength of the fail^ without augmenting the quantity of cloth, does is not increafe the power $ becaufe what is gained by the length of the lever, is loft by the flownefs of the rotation. . . Corol. 2. If fails are increafed in length, the breadth remaining the fame, thejefied will be as the radius. • . VI. Concerning the velocity of the extremities of windmill fails, in re/pe£l to the velocity of the wind. Maxim p. The velocity of the, extremities of Dutch fails, as well as of the enlarged fails, in all their ufual pqfitions when unloaded, or even loaded to a maximum^ are confderably quicker than the velocity of the wind. •* The Dutch fails unloaded, as in Tab. 3. N<> 8. made iao revolutions in fa": the diameter of the fails beiqg 3 feet 6 inches, the velocity of their ex- tremities will be $5,4 feet in a fecond ; but the velo- city of the wind producing it, being 6 feet in the fame time, we (hall have 6: 25**4: ;i 14,2 j in this cafe therefore, the velocity of their extremities was 4,2 times greater than that of the wind. In like manner, the relative velocity of the wind, to the .ex- tremities of the fame fails, when loaded to a maxi- mum, making then 93 turns in 52", will be found to 'be as x : 3,3 ; or 3,3 times quicker than that of the iwind. 5 The The, following table contains 6 examples of Dutch fails, and 4 examples of the enlarged fails, indiffer- ent, portions, but with the conftant velocity, of the wind of 6 feet in a fecond, from table 3 : and alfo 6 examples of Dutch fails in different portions, with different velocities of the wind, from table 4. Table V. containing the rath of the velocity of the extremities of windmill fails to the velocity tf tho wind. r 2 3 4 i 8 9 10- II I !3 7 8 9 10 11 £2 t *3 14 IX 10 I .' 1 ■ H ■A J 7 fms*+-m t 5 3 5 7* 19 i%- ti 10 !« «5 . 5 5 . Hi 7* 10 10 'O.S s .2 '£ 8 6 o 60 6 o 4 Q 6 o 6 o 60 60 * UIJ'l, 4 4* 8 9 4 4> 8 9 .4 4i 8.9 7WIT1WW 4* 11 Mil I III II 11 j ,1 uj, I H«PQ of we veloqty of the wind and ex- tremities of the fails. unloaded, loaded. It r 66 J ... \ It appears from the preceding colle&ion of ex- amples, that when the extremities of the Dutch fails- are parallel to the plane of motion, or at right angles- to the wind, and to the axis, as they are made accord- ing to the common practice in England> that their velocity, unloaded, is above 4 times, and loaded ta a maximum^ above 3 times greater than that of the- wind : but that when the Dutch fails, or enlarged, fails, are in their beft pofitions, their velocity un- loaded is 4 times, and loaded to a maximum, at a. taedium the Dutch fails are 2,7, and the enlarged fails 2,6 times greater than the velocity- of the wind*. Hence we are nirnifhed with, a method of knowing the velocity of the wind, from obferving the velocity of the windmill fails %. for knowing the radius, and 1 the number of turns in a minute, we fliall have the velocity of the extremities ; which, cliv^ded by the following divifors, will give the velocity \ of the. wind. Dutch fails in their common pofitidnIf nl ^ ed +- a t loaded —3.3 Dutch Oils in their beft pofition - ^f^tX^ Enlarged fails in their beft pofition {^J^ J 5 From the above divifors there arifes the following; compendiums ; fuppofing the radius to be 30 feet,, which is, the moft ufual length in this country, and the. mill to be loaded to a maximum^ as is ufuallv the. cafe with corn mills ; for every 3 turns in a mtnute y , of the Dutch falls in their common pofition^ the wind will move at the rate of 2. miles an hour $ for every 5 turns in a mtnutey of the Dutch fails in their heft " pofitim> [ 67 3 fofition, the wind moves 4 miles an hour ; and for every 6 turns in a minute > of the enlarged fails in their beji pojition , the wind will move 5 miles. an hour. The following table, which was communicated -to me by my friend Mr. Roufe, and which appears to have been conftrudted with great care, from a con- fiderable number of fads and experiments, and Which! having relation to the fubjedt of this article; I here infert it as he fent it to me ; but at the fame time muft obferve, that the evidence for .thofe numbers where the velocity of the wind exceeds jo miles an hour; do not feem of equal authority with thofe of 50 miles an hour and under. It is alfo to be obferved, that the numbers incol. 3. are calculated according to the fquare of the velocity of the wind, which, in mode- rate velocities, from what has been before obferved, will hold very nearly, t t . I z Tabl* * * <• % -4 t 68 J « » . . " ' » ... * Table VI. containing the velocity ttnd force of windy according to their common appellations. Velocity x»f the Wind. Lag "a* ^.ai i 2 3 4 5 ! ro. IS 20 »S 30 35 40 145 50 00 80 100 1 .a 8es ^ leu o •*»47 fir -22,00. 29,34 36,67 44,oi 51.34 58,68 66,01 35 ,02 «7.36 146,70 o £ * 1 appdlstk>M> ofidwfow »oos ,020 _ ,044 #079 ^3 3$ 3»075 4>4*9 6,027 7*873 9>9 6 3 12,300 i7i7 x S 131,490 49,200 it • 1 1 J iilHi l m*m**~m+«4 Hardly peftjfptiblc. |Jdfti>efCe^tiWe. , r Gentle, pleafant wind. ciPteafant brifk gale. J r » ** ... f Very brifk. 1 High winds* I Very high. A florin or tempeft, A great ftorm. An hurricane. An hurircane that tears tip trees, carries buildings before it, &c. VII* concerning the ahfolute effedl, produced by a> given velocity of the windy upon fails of a given? magnitude and conftrufiion. It has been obferved by pra&itioners, that in mills- with 'Dutch fails in theccinmon pofition, that when they make about ij turns in a minute, they then work. > . . » . •#• V T*9j -*rork itt a mean rate : that is, by the compendium* in the laft article, when the velocity of the wind is .$% miles an hour, or ia-|feetin a ieoond; vfrhich r :ia common phrafe, would be called zfrejb gale. The experiments fetdown in Tab. IV. N° 4, were ; tried with a wind, whofe velocity was 81 feet in, a? dEscond; . confequently had thofe experiments been -tried with a wind, whofe velocity was i2«f feet in:a fttond, the cfifcft, hy maxim 3d, would have been 3 times greater ; becaufe the cube of u\h 3 times greater than that of 8$. Emm Tab* IV. N04. ,we find, that the fails, v?h$ n % velocity of the wind was 8| feet in frfcconiy made rjo revolutions in a minute, with a load <?f *ZAS fc* >From tbe.raea&ces of .the, machine, pre- ceding the specimen taf&.&tt . of .experiments, wc find, Jthat 2 o revolutions pf the fails raifed, the fcafe and weight 1 u ,3 inches: 130 revolutions will there- . fore, nrife the- fcale y% >4f inches, -. which* multiplied fey *7>5 a fcr makes japrodtfft of i*&7> for the «f&^ oftbeDutdi fails in their heft pofition r that is, whoa the velocity of the wind is Si feet in a fecond : this produd therefore multiplied by ^ee,. will give 386r for the effe<£t of the fame fails, whence velocity $£ the wind is . 1.2? feet in a fecood. Defagiiliers makes the utmoft power of a m$n r when working fo as to he able to hold it for fopie hours, to be equal- to that of raifing an hogfhead of .water : to; feet h^hkuai minute. ;Nwv, * an .hog&ead* confifting of 63 ale gallons, .being redwed;inf<> pounds averdupois, and the height into inches ; the produd made by s multiplying thofe two/iwmbejrs will be 76800 j* whiqh is 19 itin^cs greater. than the ■>.,.....«.-. I 7° '■] ■fprodutt of the fails lafl>mentioned, at I'ffbet'ina •Second : therefore, by maxim 8th, if we multiply «the fquare root of 19, that is 4,46, by. 21 inches^ the-length of the fail producing the effe6t 3861, ws fhall have 93,66 inches, or 7 feet 94 inches for the radius of a Dutch fail in its beft pofition, whbfe mean -power (hall be equal to that of a man : but if they are -in their common pofition, their length muft be in- ^reafed inthe ratio of the fquare root of 442 to that -of 639, as thus appears j ! The ratio of the maximum produftsi of ffi°- 8 and «ii. Tab. IIL are as 442:639 ; but by maxim 8, the effects of fails of different radii are as the fquaife <if the radii ; confequently the the fquare roots of the products or effe&s, are as the radii fimply : ; and -therefore as the fquare root of 44a is to that of ^39 > f° is 93,66 to 112 j66', or 9 feet 4| inches. . If the fails are of the enlarged kind, then from Tab. IIL N° 1 rand 15. we -fliall have the fquare *oot ef 8ao tovthatof 4539; : 93,66 : 82,8 inches, or 6 feet 1 0$ inches : fo that in round numbers we fliall have the radius of a fail, of a fimilar figure to their refpedtive models, whofe mean power fhall be equal -to'tfhat of a man ; The Dutch fails in their common' pofition 9! feet. The Dutch /ails in their beft pofition — - 8 The enlarged fails, ih their beft pofition — 7 Suppofe now the radius of a* fail to.be 30 feet, and 4o be conftru^d * upon the model of the enlarged -fails, N° 14 or 15. Tab.'III. .diyiding 30 by 7<wje, HQiali have 4,28, the fquare of which is 18,3; and r £his, jaccotding to maxim 7, will be the relative pow^r [ 7« ] power of a foil of jo feet, to one of 7 feet ; that is* when working at a mean rate, the 30 feet fail will be equal' to the power of 18^3 men, or of 3-|-horfes ; reckoning 5 men to a. horfe : whereas the efFed of the common Dutch fails, of the fame length, being lefs in the proportion of 820:442,, will be fcarce equal to the power of 10 men,, or of 2 horfea. That thefe computations, are not merely fpecula- tive* but will nearly hold good when applied to works in large,. I have had an opportunity of verify- ing : for in a- mill with the enlarged fails of 30 feet, applied to the crufhingof rape feed, by means of two runners upon the edge, for making. oil; I oh— fcrved r that when the fails made n turns in a mi- nute, in which cafe the velocity of the wind was* about 13 feet in a fecond, according to article 6th, that the runners then made 7 turns in a minuter whereas 2 horfes, applied, to the fame ^ runners, fcarcely* worked them at the rate of 3.1 turns in, the fame time. Laftly,, with regard to the real fuperio- rity of the enlarged fails, above the Dutch fails as commonly made r it has fufficiently appeared, not only in thofe cafes where they have been applied to new mills, but where they have been fubuituted in the place of the others. yill. Concerning horizontal windmills and water- wheels, with oblique vanes.. Obfervations upon the~effeds of common wind- mills with oblique vanes, have led many to imagine, that could the vanes be brought to receive the direct impulfe, like a ihip failing before the wind, it would fce a rery great improvement i& point of power : while ofehers attending to the extraordinary and even unexpected effefts of oblique vanes, have Been led to imagine, that oblique vanes applied to wtfter-miUe, would as much exceed the common water wheels* as the vertical windmills are found to have exceeded all attempts towards an horizontal one. Both theffe notidns, but efpecially the firft, have fo plaufiHe an appearahce, that of late yeaft there has fcldom beeri Wanting thdfe, who have affiduoufly employed fhem- felvesto bring ttt bear deftgrts of thkkind: it may not therefore' be unacceptable to endeavour to fet this ttiatttr in a clear light. * Plate VI. £g 2d. Let AB be the &&*ow of a plain, upbn which let the vyittd blow in the dire&ieti C D, tvith fiich a velbeity as to deferibe a given fpace U % in a given time (ffcppbffe i feeond) ; and let A B be mewed parallel to itfetf, in the direc- tion C D. Ho W, if the plane A B moves' with tfefc lame velocity *s the Wind j that is, if the point B ihdvts thto' tfcefpate^E in the feme tit^e tfcat a particle of ait wt^tdd move thro' the feme fp&ee 5 k as plain that, hi this cafe, AeteCan be no preffureor iirhptrife of the wind upon the plane : but if tfee plane moves flower than the wind, in the- feme efire&ka*, fo that the point B may move to. F, while a particle *>f air, fetthig out ftttfn B at the feme mftaht, would move to E, then B F wSl eiprfcfe the velocity cif the plane; and the relative velocity of the wind and plane Will be exprfefibd by the iineVE. Let the ra«fe of • f E to BE be giVeh (ftfppofe 2 r 3.) i let the lint 'A B rep rfefent thfe Trtipulfe of Ac vmA upon Ae plane A B, Wheh 'a&irtg'With its:i»htofle vetoeity B E 3 but* «* •* \ I ) 1 C 73 3 * • * when adting with its relative velocity F E, let its im- pulfe be denoted by fome aliquot part of A B, as for inftance 4 A B : then will -J of the parallelogram A F reprefent the mechanical power of the plane $ that is* ^ ABx^BE. idly, Let IN be the fedtion of a plane, inclined in fuch a manner, that the bafe IK of the redtangle triangle I K N may be equal to A B ; and the per- pendicular N K=B E ; let the plane IN be ftruck by! the wind, in the diredtion L M, perpendicular to I K : then, according to the known rules of oblique forces, the impulfe of the wind upon the plain I N, tending to move it according to the diredtion L M, or NK, will be denoted by the bafe IKj and that part of the impulfe, tending to move it according to the diredtion I K, will be expreffed by the perpendi- cular N K. Let the plane I N be moveable in the diredtion of I K only } that is, the point I in the di- redtion of I K, and the point N in the diredtion N (X parallel thereto. Now it is evident, that if the point I moves thro' the line I K, while a particle of air," fetting forwards at the fame time from the point N, moves thro' the line N K, they will both arrive at the point K at the fame time; and confequently, in this cafe alfo, there can be no preflure or impulfe of the particle of the air upon the plane I N. Now let I O be to I K as B F to B E ; and let the phtne I N move at fuch a rate, that the point I may arrive at O, and acquire the pofitioh I Q^in the fame time that a par- ticle of wind would move thro* the ipace N K : as O QJs parallel to IN; (by the properties erf fimilar triangles) it will cut N K in the point P, in fuch a #»anner, that N P=B F* and P fc^F E i -hence it K appears, 1 t74] appears, that the plane I N, by acquiring the pofi* tion O (^withdraws itfelf from the a&ion of the : wind, by the fame fpace N P, that the plane A B does by acquiring the pofition FG $ and consequently, from the equality of P K to F E, the relative im- pulfe of the wind P K, upon the plane. O Q, will be equal to the relative impulfe of the wind F E, upon the plane F G : and fince the impulfe of the wind upon AB, with the relative velocity FE, in the di- rection B E, is reprefented by 4 A B ; the relative impulfe of the wind upon the plane I N, in the di- rection NK, will in like manner be reprefented by •| I K ; and the impulfe of the wind upon the plane I N, with the relative velocity P K* in the direction * I K, will be reprefented by ^ N K : and confequently the mechanical power of the plane I N, in the direc- tion I K, will be 4 the parallellograro I Qj that is •j I JC x I N K : that is, from the equality *>f I KraAB*; and N K=B E, we foall have f I Q=| ABx|fi£. =4A B x £ B E=4 of the area of the paraUdlogom * A F. Hence we deduoe this General Proposition, That all planes, however fkuated y jhat intercept* the fame feSlion of the wind, and having the fame re- - Iqtive velocity y in regard to the windy when reduced < into the fame dire£tion> have equal powers to produce mechanical effe&s. For what is loft by the obliquity of the impulfe, is gained by the velocity of the motion. \ Hence it appears, that an oblique M is under no ■> difadvantage in refpedt of power, compared with a \ direct one j except what arifes from a diminution of it** C75] it* breadth, ip refpftft to the fz&wn of (he wind : 4hp breadth J N teiog by obliquity reduced *o I K. The difadyaottge of horizontal windmills there- fore docs not confift in this ; th«t each fail, when .dire&ty expofed to the wind, is capable of a lefs jpower, than zn oblique one of the fame dhnenfions ; jbuttha(t in m Jhorizqnfal windmill* little more than one fail c^n he a&ing at. once : whereas in the com^ moa windmill, all the fonjr ,a#: together : a&d therer fore, Aippofijig e»ch vane of a$ horizontal windmill, c£ the fiuaae -dimensions as each vane of the vertical, it is jj^anifeft the power of a vertical njill with four fails, wilj he /our tinaes greater than the power of the horizontal owe, let its number of vanes be what it will ; this 4ifa4vairtage arifes from jthe nature of the thiogj but if we confid$r tfce further difedwaage, thajtoarifes from tfte difficulty of getting the fails back again ^g^A the *?ind, &q. we *eed n<& wooder if this Jdnd of wiH is in reality found to have not above jorrj pf $he power of the common fort ; as ha* ap- peared in fame attempts of this kifld. In like manner, as little improvement is to be ex- pected from water -mills withidbKque vanes : for the power of the iamefe&ipQ of a stream of water, is not gceater when acting upon an ^oblique vane, ithan whe* aaing.^p«i a dWt one: and »ny advantage thatch he faade by intercepting 3 >gj?eater ft&ion, isfftiqh ^m^tiroesrit^fee doije in the, cafe -of w open xwer, wiU be ^aqnteibalJanced by tthe fupecior p efift> ajace, jhsit % h v*pfi6 dffpuid snee.t withiby impviijng.at *ight ^ngjqs $i> $hg cjirrent : whereas the qamoioo &¥*tft 4s^s'J»S¥e Wjiihifhe iWfttpriieady in&c fame K 2 Here [ 76 ] Eferc it may reafbnably be afkcd, that fihce our geometrical demonftration is general, and proves, that one angle of obliquity is as good as another ; why in our experiments it appears, that there is a certain angle which is to be preferred to v all the reft? It is to be obferved, that if the breadth of the fail I N is given i the greater the angle KIN, and the lefs will be the bafe I K : that is, the fe&ion of wind inter- fered, will be lefs : on the other hand, the more acute the angle KIN, the lefs will be the perpendi* cular KN: that is, the impulfe of the wind, in the direction I K being lefs, and the velocity of the fail greater ? the refiftance of the medium will be greater alfo. Hence therefore, as there is a diminution of the fedtion of the wind: intercepted on one hand, and ^n increase of refiftance on the other, there is fome angles where the difadvantage arifing from thefe- caufeis up* on the whole is the leaft of all ; but as thfe difadvan- tage arifing from refiftance is more of a phyfical than geometrical confideration* the true angle will beft be affigned by experiment SC HO LI UM.. In trying the experiments contained in Tab. Ill; and IV. the different fpecific gravity of the air, which is undoubtedly different at different times, will caufe a difference in the load, proportional to the difference of its fpecific gravity, tho* its velocity remains the fame ; and a variation of fpecific gravity may arife not only from a variation of the weight of the wholef column, 1 but alfo by the difference of heat of the air concerned in the experiment, and poffibly of other caufes s yet the irregularities that might arife from & di£- C 77 ] difference of fpecific gravity were thought tb Be too (mall to be perceivable, till after the principal experiments were made, and* their effedts compared ;: from which, as well as fucceeding experiments, thofe variations were found to- be capable of producing a fenfible, tho' no very confiderable efied; : however, as all the experiments were tried in the fummer fea- ibn, in the day-time, and under cover ; we may fup- pofe that the principal fource of error would arife from the different weight of the column of the atmo- fphere at different times; but as this feldom varies above T t part of the whole, we may conclude, that tho' many of the irregularities contained in the experi- ments referred to in the foregoing eflay, might arife from this caufe ; yet as all the principal conclufions are drawn from the medium of a confiderable num- ber, many whereof were made at different times, it is prefumed that they will nearly agree with the truth, and be altogether fufficient for regulating the pra&ical conftru&ion of thofe kind of machines, for which ufe they were principally intended* V '' mm If. <- . y> •> / r>- i t-ty* i/: - I i »» V. .».- r : • •, * <> v ^ ^•^*- I \/ 1 . v ? • • • *